Equation as terrain
Mordell-style analysis begins by making the equation navigable: normalize coefficients, record bad primes, test residues, separate rational from integral solutions, and decide whether a descent or parametrization is plausible.
00
Reconstruction Method
This page is a bibliographic and methodological reconstruction, not a reproduction of Mordell’s books or papers. Strategy tags overlap: one case can instantiate several methods, so prevalence percentages indicate case usage rather than a probability distribution.
Descent and basis
The finite-basis theorem reframes rational points on cubics as a finitely generated arithmetic group. The working algorithm is not merely solving equations; it is finding a finite generator set behind infinitely many rational points.
Problem ecology
Mordell’s mathematics often turns hard concrete examples into families, then into chapters, conjectures, and problem lists. The lasting object is a research ecology: local tests, forms, curves, examples, and open directions.
01
33 Mordell Strategies
A
Diophantine obstruction
M0115.7% · 47/300 cases
Turn equation into arithmetic terrain
\(F(x_1,\ldots,x_n)=0\Rightarrow\{\text{local},\text{global},\text{infinite}\}\)
Refuse to treat a Diophantine equation as a bare polynomial; treat it as a terrain with congruence shadows, rational parametrizations, descent paths, and exceptional points.
M0213.7% · 41/300 cases
Begin with congruence impossibility
\(F(\mathbf x)\not\equiv 0\pmod m\)
Before attempting construction, test whether residues already forbid solutions; the smallest modulus may contain the entire theorem.
M0312.7% · 38/300 cases
Separate integral from rational solvability
\(X(\mathbb Z)\subset X(\mathbb Q)\)
Distinguish rational parametrization from integer representation; a rational point may create structure while integral points require arithmetic control.
M0410.3% · 31/300 cases
Normalize coefficients and discriminants
\(F\sim F'\quad(\Delta(F')\ \text{controlled})\)
Simplify an equation by transformations that preserve the arithmetic question but expose discriminants, contents, and bad primes.
M0512.0% · 36/300 cases
Use descent as arithmetic compression
\(P\mapsto P',\quad h(P')
Replace an infinite search by a strictly decreasing height, norm, denominator, or form invariant.
M067.3% · 22/300 cases
Exploit local-to-global failure
\(X(\mathbb Q_p)\ \forall p\not\Rightarrow X(\mathbb Q)\)
Use the mismatch between local data and global solutions as a diagnostic signal, not merely as an obstacle.
B
Elliptic curves and rational points
M0715.0% · 45/300 cases
Make a cubic into a group
\(E(\mathbb Q): y^2=x^3+ax+b\)
Turn a plane cubic with a rational point into an arithmetic group whose addition law converts geometry into algebra.
M0814.0% · 42/300 cases
Prove finite generation by finite basis
\(E(\mathbb Q)\cong \mathbb Z^r\oplus E(\mathbb Q)_{\rm tors}\)
Search for a finite basis of rational points, then make every rational point a finite combination of generators.
M0911.0% · 33/300 cases
Use heights to control rational points
\(\hat h(nP)=n^2\hat h(P)\)
Convert arithmetic complexity into a numerical height that descends, grows quadratically, or bounds generators.
M1013.3% · 40/300 cases
Study Mordell curves as laboratories
\(y^2=x^3+k\)
Use the family now called Mordell curves as a controlled model for cubic Diophantine phenomena.
M119.3% · 28/300 cases
Distinguish torsion from rank
\(E(\mathbb Q)=E_{\rm tors}\oplus E_{\rm free}\)
Separate finite exceptional structure from the free part that encodes infinite rational generation.
M128.7% · 26/300 cases
Promote examples into conjectures
\(\#X(\mathbb Q)<\infty\quad(g>1)\)
Let stubborn patterns in rational points become explicit conjectures, especially the finiteness principle later called the Mordell conjecture.
C
Forms and representations
M1313.0% · 39/300 cases
Reduce equations to binary forms
\(F(x,y)=m\)
Transform Diophantine equations into questions about binary quadratic, cubic, or quartic forms and their equivalence classes.
M148.0% · 24/300 cases
Use composition of forms
\([f]\cdot[g]=[h]\)
Exploit the algebra of equivalence classes of forms so that representation becomes multiplicative structure.
M159.0% · 27/300 cases
Read ideals through forms
\(\mathfrak a\leftrightarrow [ax^2+bxy+cy^2]\)
Translate between algebraic number theory and classical forms to import factorization, norms, and class information.
M168.3% · 25/300 cases
Use geometry of numbers bounds
\(\Lambda\cap K\ne\varnothing\)
Turn arithmetical existence into lattice-point existence inside a convex region or bounded domain.
M179.7% · 29/300 cases
Control exceptional representations
\(R_F(m)=\#\{(x,y):F(x,y)=m\}\)
Do not merely prove representability; count, classify, and isolate exceptional representations.
M187.7% · 23/300 cases
Parametrize before counting
\(X\dashrightarrow \mathbb P^1\)
When genus-zero or rational structure appears, parametrize first, then impose integrality or coprimality.
D
Modular forms and analytic tools
M1910.7% · 32/300 cases
Use modular identities as arithmetic engines
\(\sum_{n\ge1}\tau(n)q^n\)
Treat a modular identity as a machine producing congruences, multiplicativity, and arithmetic coefficients.
M207.0% · 21/300 cases
Prove multiplicativity from structure
\(\tau(mn)=\tau(m)\tau(n)\quad((m,n)=1)\)
Show that a sequence is not a list of accidents but a multiplicative arithmetic object.
M216.0% · 18/300 cases
Convert sums into transforms
\(S(a,b)=\sum_n e^{2\pi i f(n)}\)
Re-express hard arithmetic sums through transformation formulas, reciprocity, or analytic continuation.
M226.7% · 20/300 cases
Extract congruences from generating functions
\(A(q)\equiv B(q)\pmod p\)
Let formal series produce residue information and divisibility constraints for coefficients.
M2310.0% · 30/300 cases
Mix elementary and analytic methods
\(\text{elementary descent}+\text{analytic estimate}\)
Refuse a false dichotomy between elementary number theory and analysis; combine whichever controls the invariant.
M245.3% · 16/300 cases
Use asymptotic evidence cautiously
\(N(B)\sim C B^\alpha(\log B)^\beta\)
Let asymptotics guide conjecture formation while keeping exact Diophantine assertions separate.
E
Problem ecology and exposition
M2511.7% · 35/300 cases
Make a problem family the unit
\(\{F_t(\mathbf x)=0\}_{t\in T}\)
Study not one equation but a family whose parameters reveal what changes and what remains invariant.
M2611.3% · 34/300 cases
Collect problems as research infrastructure
\(\text{examples}\to\text{problems}\to\text{theorems}\)
Use problem lists, examples, and marginal questions as a durable machine for producing new theorems.
M2710.0% · 30/300 cases
Write discursively but compute concretely
\(\text{narrative}\Rightarrow\text{worked arithmetic}\)
Move between informal explanation and explicit calculations; make the reader see the arithmetic by doing it.
M288.3% · 25/300 cases
Use book chapters as theorem maps
\(\text{chapter}\simeq\text{method atlas}\)
Let a monograph chapter organize a subfield by methods, examples, exceptions, and historical dependencies.
M296.3% · 19/300 cases
Preserve historical priority
\(\text{idea}\mapsto\text{source}\mapsto\text{variant}\)
Treat provenance, priority, and variant methods as mathematical information rather than literary ornament.
M305.7% · 17/300 cases
Build a school through problems
\(\text{seminar}+\text{visitors}+\text{problems}\Rightarrow\text{field}\)
Create a research ecology where young mathematicians and visitors inherit hard concrete problems.
F
Meta-strategies
M3114.7% · 44/300 cases
Prefer hard concrete instances
\(\text{specific }k\Rightarrow\text{general pattern}\)
Attack a stubborn special case until it reveals the invariant behind a general theorem.
M3210.3% · 31/300 cases
Turn impossibility into structure
\(\varnothing\ne\text{proof of no solution}\)
A nonexistence proof is not a dead end; it records congruence, descent, or class-group structure.
M3312.3% · 37/300 cases
Let arithmetic taste guide abstraction
\(\text{calculation}\leadsto\text{concept}\)
Allow repeated computation and number-theoretic taste to select the abstraction only after the examples demand it.
02
Overlapping Prevalence Ranking
Bars show reconstructed case prevalence out of 300. The counts are intentionally overlapping because a Mordell-style proof may simultaneously use congruences, forms, descent, and problem-family reasoning.
03
300-Case Corpus
300/300 shown
| # | Years | Family | Case | Methodological thesis | Tags |
|---|---|---|---|---|---|
| 001 | 1912-1918 | Early Diophantine equations | Congruence tests for cubic equations | Treat congruence tests for cubic equations as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M01M02M04 |
| 002 | 1912-1918 | Early Diophantine equations | Quadratic reciprocity in cubic problems | Use quadratic reciprocity in cubic problems to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M02M04M05 |
| 003 | 1912-1918 | Early Diophantine equations | Ideal-number attack on early cubics | Read ideal-number attack on early cubics as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M04M05M31 |
| 004 | 1912-1918 | Early Diophantine equations | Binary cubic form reduction | Reconstruct binary cubic form reduction through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M05M31M32 |
| 005 | 1912-1918 | Early Diophantine equations | Integral points on special cubic curves | Treat integral points on special cubic curves as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M31M32M01 |
| 006 | 1912-1918 | Early Diophantine equations | Denominator clearing and primitive triples | Use denominator clearing and primitive triples to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M32M01M02 |
| 007 | 1912-1918 | Early Diophantine equations | Residue obstructions for small moduli | Read residue obstructions for small moduli as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M01M02M04 |
| 008 | 1912-1918 | Early Diophantine equations | Parametric families of integer solutions | Reconstruct parametric families of integer solutions through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M02M04M05 |
| 009 | 1912-1918 | Early Diophantine equations | Quartic obstruction examples | Treat quartic obstruction examples as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M04M05M31 |
| 010 | 1912-1918 | Early Diophantine equations | Finite search after normalization | Use finite search after normalization to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M05M31M32 |
| 011 | 1912-1918 | Early Diophantine equations | Exceptional constants in cubic equations | Read exceptional constants in cubic equations as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M31M32M01 |
| 012 | 1912-1918 | Early Diophantine equations | Comparison of three solution methods | Reconstruct comparison of three solution methods through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M32M01M02 |
| 013 | 1912-1918 | Early Diophantine equations | Early Mordell equation prototypes | Treat early mordell equation prototypes as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M01M02M04 |
| 014 | 1912-1918 | Early Diophantine equations | Local constraints before descent | Use local constraints before descent to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M02M04M05 |
| 015 | 1912-1918 | Early Diophantine equations | Hand calculation as theorem probe | Read hand calculation as theorem probe as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M04M05M31 |
| 016 | 1912-1918 | Early Diophantine equations | Irreducibility and factorization tests | Reconstruct irreducibility and factorization tests through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M05M31M32 |
| 017 | 1912-1918 | Early Diophantine equations | Explicit impossible equations | Treat explicit impossible equations as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M31M32M01 |
| 018 | 1912-1918 | Early Diophantine equations | Arithmetic of small discriminants | Use arithmetic of small discriminants to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M32M01M02 |
| 019 | 1912-1918 | Early Diophantine equations | Specialization of a parameter | Read specialization of a parameter as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M01M02M04 |
| 020 | 1912-1918 | Early Diophantine equations | Forms behind y^2-k=x^3 | Reconstruct forms behind y^2-k=x^3 through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M02M04M05 |
| 021 | 1912-1918 | Early Diophantine equations | Congruence classes of solutions | Treat congruence classes of solutions as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M04M05M31 |
| 022 | 1912-1918 | Early Diophantine equations | Auxiliary variables and simplification | Use auxiliary variables and simplification to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M05M31M32 |
| 023 | 1912-1918 | Early Diophantine equations | From example to family | Read from example to family as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M31M32M01 |
| 024 | 1912-1918 | Early Diophantine equations | Early Diophantine notebooks | Reconstruct early diophantine notebooks through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M32M01M02 |
| 025 | 1912-1918 | Early Diophantine equations | Congruence tests for cubic equations — variation 2 | Treat congruence tests for cubic equations as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M01M02M04 |
| 026 | 1919-1925 | Finite basis theorem and elliptic curves | Chord-and-tangent group law on a cubic | Treat chord-and-tangent group law on a cubic as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M07M08M09 |
| 027 | 1919-1925 | Finite basis theorem and elliptic curves | Finite basis theorem for rational points | Use finite basis theorem for rational points to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M08M09M10 |
| 028 | 1919-1925 | Finite basis theorem and elliptic curves | Height descent on elliptic curves | Read height descent on elliptic curves as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M09M10M11 |
| 029 | 1919-1925 | Finite basis theorem and elliptic curves | Reduction of rational points modulo primes | Reconstruct reduction of rational points modulo primes through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M10M11M12 |
| 030 | 1919-1925 | Finite basis theorem and elliptic curves | Independent generators and relations | Treat independent generators and relations as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M11M12M07 |
| 031 | 1919-1925 | Finite basis theorem and elliptic curves | Torsion points versus infinite order | Use torsion points versus infinite order to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M12M07M08 |
| 032 | 1919-1925 | Finite basis theorem and elliptic curves | Cubic curves with rational base point | Read cubic curves with rational base point as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M07M08M09 |
| 033 | 1919-1925 | Finite basis theorem and elliptic curves | Mordell-Weil finite generation prototype | Reconstruct mordell-weil finite generation prototype through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M08M09M10 |
| 034 | 1919-1925 | Finite basis theorem and elliptic curves | Genus-one arithmetic as group theory | Treat genus-one arithmetic as group theory as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M09M10M11 |
| 035 | 1919-1925 | Finite basis theorem and elliptic curves | Exceptional rational points | Use exceptional rational points to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M10M11M12 |
| 036 | 1919-1925 | Finite basis theorem and elliptic curves | Infinite families from one point | Read infinite families from one point as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M11M12M07 |
| 037 | 1919-1925 | Finite basis theorem and elliptic curves | Denominator descent | Reconstruct denominator descent through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M12M07M08 |
| 038 | 1919-1925 | Finite basis theorem and elliptic curves | Basis search by height | Treat basis search by height as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M07M08M09 |
| 039 | 1919-1925 | Finite basis theorem and elliptic curves | Parametrized secant construction | Use parametrized secant construction to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M08M09M10 |
| 040 | 1919-1925 | Finite basis theorem and elliptic curves | Finite index subgroup argument | Read finite index subgroup argument as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M09M10M11 |
| 041 | 1919-1925 | Finite basis theorem and elliptic curves | Comparison with Poincare's question | Reconstruct comparison with poincare's question through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M10M11M12 |
| 042 | 1919-1925 | Finite basis theorem and elliptic curves | Conjectural finiteness for genus greater than one | Treat conjectural finiteness for genus greater than one as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M11M12M07 |
| 043 | 1919-1925 | Finite basis theorem and elliptic curves | Rational solutions from tangent lines | Use rational solutions from tangent lines to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M12M07M08 |
| 044 | 1919-1925 | Finite basis theorem and elliptic curves | Failure modes in naive parametrization | Read failure modes in naive parametrization as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M07M08M09 |
| 045 | 1919-1925 | Finite basis theorem and elliptic curves | Elliptic curve arithmetic tables | Reconstruct elliptic curve arithmetic tables through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M08M09M10 |
| 046 | 1919-1925 | Finite basis theorem and elliptic curves | Birational changes of a cubic | Treat birational changes of a cubic as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M09M10M11 |
| 047 | 1919-1925 | Finite basis theorem and elliptic curves | Rank evidence from examples | Use rank evidence from examples to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M10M11M12 |
| 048 | 1919-1925 | Finite basis theorem and elliptic curves | Group law as Diophantine machine | Read group law as diophantine machine as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M11M12M07 |
| 049 | 1919-1925 | Finite basis theorem and elliptic curves | From theorem to conjecture | Reconstruct from theorem to conjecture through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M12M07M08 |
| 050 | 1919-1925 | Finite basis theorem and elliptic curves | Chord-and-tangent group law on a cubic — variation 2 | Treat chord-and-tangent group law on a cubic as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M07M08M09 |
| 051 | 1914-1969 | Mordell curves and cubic models | The equation y^2=x^3+k | Treat the equation y^2=x^3+k as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M10M01M03 |
| 052 | 1914-1969 | Mordell curves and cubic models | Integral points on Mordell curves | Use integral points on mordell curves to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M01M03M05 |
| 053 | 1914-1969 | Mordell curves and cubic models | Cubic discriminant control | Read cubic discriminant control as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M03M05M17 |
| 054 | 1914-1969 | Mordell curves and cubic models | Twists and special constants | Reconstruct twists and special constants through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M05M17M31 |
| 055 | 1914-1969 | Mordell curves and cubic models | Small k as research laboratory | Treat small k as research laboratory as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M17M31M10 |
| 056 | 1914-1969 | Mordell curves and cubic models | Descent on y^2=x^3+k | Use descent on y^2=x^3+k to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M31M10M01 |
| 057 | 1914-1969 | Mordell curves and cubic models | Rational versus integral solutions | Read rational versus integral solutions as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M10M01M03 |
| 058 | 1914-1969 | Mordell curves and cubic models | Congruence sieve for Mordell curves | Reconstruct congruence sieve for mordell curves through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M01M03M05 |
| 059 | 1914-1969 | Mordell curves and cubic models | Factorization in quadratic fields | Treat factorization in quadratic fields as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M03M05M17 |
| 060 | 1914-1969 | Mordell curves and cubic models | Exceptional integral points | Use exceptional integral points to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M05M17M31 |
| 061 | 1914-1969 | Mordell curves and cubic models | Comparison of k values | Read comparison of k values as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M17M31M10 |
| 062 | 1914-1969 | Mordell curves and cubic models | Cubic form viewpoint | Reconstruct cubic form viewpoint through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M31M10M01 |
| 063 | 1914-1969 | Mordell curves and cubic models | Height bounds for special curves | Treat height bounds for special curves as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M10M01M03 |
| 064 | 1914-1969 | Mordell curves and cubic models | Reduction to primitive solutions | Use reduction to primitive solutions to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M01M03M05 |
| 065 | 1914-1969 | Mordell curves and cubic models | Parametric subfamilies | Read parametric subfamilies as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M03M05M17 |
| 066 | 1914-1969 | Mordell curves and cubic models | Class-group obstructions | Reconstruct class-group obstructions through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M05M17M31 |
| 067 | 1914-1969 | Mordell curves and cubic models | Positive and negative k | Treat positive and negative k as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M17M31M10 |
| 068 | 1914-1969 | Mordell curves and cubic models | Search tables for solutions | Use search tables for solutions to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M31M10M01 |
| 069 | 1914-1969 | Mordell curves and cubic models | Mordell equation in lectures | Read mordell equation in lectures as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M10M01M03 |
| 070 | 1914-1969 | Mordell curves and cubic models | Generalized cubic equations | Reconstruct generalized cubic equations through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M01M03M05 |
| 071 | 1914-1969 | Mordell curves and cubic models | Cubics as finite-basis examples | Treat cubics as finite-basis examples as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M03M05M17 |
| 072 | 1914-1969 | Mordell curves and cubic models | Historical variants of the equation | Use historical variants of the equation to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M05M17M31 |
| 073 | 1914-1969 | Mordell curves and cubic models | Computational reconstruction of cases | Read computational reconstruction of cases as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M17M31M10 |
| 074 | 1914-1969 | Mordell curves and cubic models | Problem-list extensions | Reconstruct problem-list extensions through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M31M10M01 |
| 075 | 1914-1969 | Mordell curves and cubic models | The equation y^2=x^3+k — variation 2 | Treat the equation y^2=x^3+k as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M10M01M03 |
| 076 | 1920-1960 | Binary forms and representation | Binary quadratic representation | Treat binary quadratic representation as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M13M14M15 |
| 077 | 1920-1960 | Binary forms and representation | Binary cubic forms | Use binary cubic forms to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M14M15M16 |
| 078 | 1920-1960 | Binary forms and representation | Binary quartic forms | Read binary quartic forms as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M15M16M17 |
| 079 | 1920-1960 | Binary forms and representation | Equivalence of forms | Reconstruct equivalence of forms through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M16M17M18 |
| 080 | 1920-1960 | Binary forms and representation | Composition and representation | Treat composition and representation as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M17M18M13 |
| 081 | 1920-1960 | Binary forms and representation | Discriminant stratification | Use discriminant stratification to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M18M13M14 |
| 082 | 1920-1960 | Binary forms and representation | Class-number style constraints | Read class-number style constraints as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M13M14M15 |
| 083 | 1920-1960 | Binary forms and representation | Representation by homogeneous forms | Reconstruct representation by homogeneous forms through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M14M15M16 |
| 084 | 1920-1960 | Binary forms and representation | Thue equation reductions | Treat thue equation reductions as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M15M16M17 |
| 085 | 1920-1960 | Binary forms and representation | Special binary forms with finite solutions | Use special binary forms with finite solutions to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M16M17M18 |
| 086 | 1920-1960 | Binary forms and representation | Forms from algebraic integers | Read forms from algebraic integers as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M17M18M13 |
| 087 | 1920-1960 | Binary forms and representation | Ideal classes and forms | Reconstruct ideal classes and forms through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M18M13M14 |
| 088 | 1920-1960 | Binary forms and representation | Parametrizing rational conics | Treat parametrizing rational conics as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M13M14M15 |
| 089 | 1920-1960 | Binary forms and representation | Ternary form reductions | Use ternary form reductions to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M14M15M16 |
| 090 | 1920-1960 | Binary forms and representation | Quaternary quadratic forms | Read quaternary quadratic forms as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M15M16M17 |
| 091 | 1920-1960 | Binary forms and representation | Counting representations | Reconstruct counting representations through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M16M17M18 |
| 092 | 1920-1960 | Binary forms and representation | Exceptional representation classes | Treat exceptional representation classes as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M17M18M13 |
| 093 | 1920-1960 | Binary forms and representation | Minkowski-style bounds | Use minkowski-style bounds to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M18M13M14 |
| 094 | 1920-1960 | Binary forms and representation | Primitive representation constraints | Read primitive representation constraints as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M13M14M15 |
| 095 | 1920-1960 | Binary forms and representation | Reduction of coefficients | Reconstruct reduction of coefficients through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M14M15M16 |
| 096 | 1920-1960 | Binary forms and representation | Local representation tests | Treat local representation tests as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M15M16M17 |
| 097 | 1920-1960 | Binary forms and representation | Global representation failures | Use global representation failures to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M16M17M18 |
| 098 | 1920-1960 | Binary forms and representation | Forms in Diophantine Equations | Read forms in diophantine equations as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M17M18M13 |
| 099 | 1920-1960 | Binary forms and representation | Historical comparison of form methods | Reconstruct historical comparison of form methods through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M18M13M14 |
| 100 | 1920-1960 | Binary forms and representation | Binary quadratic representation — variation 2 | Treat binary quadratic representation as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M13M14M15 |
| 101 | 1917-1930 | Modular forms and Ramanujan tau | Multiplicativity of Ramanujan's tau | Treat multiplicativity of ramanujan's tau as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M19M20M22 |
| 102 | 1917-1930 | Modular forms and Ramanujan tau | q-series as coefficient machine | Use q-series as coefficient machine to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M20M22M23 |
| 103 | 1917-1930 | Modular forms and Ramanujan tau | Modular identities in arithmetic | Read modular identities in arithmetic as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M22M23M24 |
| 104 | 1917-1930 | Modular forms and Ramanujan tau | Congruences for coefficients | Reconstruct congruences for coefficients through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M23M24M29 |
| 105 | 1917-1930 | Modular forms and Ramanujan tau | Ramanujan-style experimental evidence | Treat ramanujan-style experimental evidence as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M24M29M19 |
| 106 | 1917-1930 | Modular forms and Ramanujan tau | Products and generating functions | Use products and generating functions to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M29M19M20 |
| 107 | 1917-1930 | Modular forms and Ramanujan tau | Euler products and multiplicativity | Read euler products and multiplicativity as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M19M20M22 |
| 108 | 1917-1930 | Modular forms and Ramanujan tau | Transformation formula evidence | Reconstruct transformation formula evidence through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M20M22M23 |
| 109 | 1917-1930 | Modular forms and Ramanujan tau | Coefficient comparison | Treat coefficient comparison as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M22M23M24 |
| 110 | 1917-1930 | Modular forms and Ramanujan tau | Arithmetic functions from series | Use arithmetic functions from series to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M23M24M29 |
| 111 | 1917-1930 | Modular forms and Ramanujan tau | Formal power series congruences | Read formal power series congruences as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M24M29M19 |
| 112 | 1917-1930 | Modular forms and Ramanujan tau | Proof from modular structure | Reconstruct proof from modular structure through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M29M19M20 |
| 113 | 1917-1930 | Modular forms and Ramanujan tau | Divisor-sum analogies | Treat divisor-sum analogies as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M19M20M22 |
| 114 | 1917-1930 | Modular forms and Ramanujan tau | Exact versus heuristic coefficient laws | Use exact versus heuristic coefficient laws to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M20M22M23 |
| 115 | 1917-1930 | Modular forms and Ramanujan tau | Tau-function history | Read tau-function history as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M22M23M24 |
| 116 | 1917-1930 | Modular forms and Ramanujan tau | Ramanujan conjectural atmosphere | Reconstruct ramanujan conjectural atmosphere through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M23M24M29 |
| 117 | 1917-1930 | Modular forms and Ramanujan tau | Modular arithmetic of series | Treat modular arithmetic of series as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M24M29M19 |
| 118 | 1917-1930 | Modular forms and Ramanujan tau | Analytic confirmation of algebraic pattern | Use analytic confirmation of algebraic pattern to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M29M19M20 |
| 119 | 1917-1930 | Modular forms and Ramanujan tau | Series identities as theorem engines | Read series identities as theorem engines as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M19M20M22 |
| 120 | 1917-1930 | Modular forms and Ramanujan tau | Coefficient tables as evidence | Reconstruct coefficient tables as evidence through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M20M22M23 |
| 121 | 1917-1930 | Modular forms and Ramanujan tau | Multiplicativity of Ramanujan's tau — variation 2 | Treat multiplicativity of ramanujan's tau as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M22M23M24 |
| 122 | 1917-1930 | Modular forms and Ramanujan tau | q-series as coefficient machine — variation 2 | Use q-series as coefficient machine to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M23M24M29 |
| 123 | 1917-1930 | Modular forms and Ramanujan tau | Modular identities in arithmetic — variation 2 | Read modular identities in arithmetic as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M24M29M19 |
| 124 | 1917-1930 | Modular forms and Ramanujan tau | Congruences for coefficients — variation 2 | Reconstruct congruences for coefficients through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M29M19M20 |
| 125 | 1917-1930 | Modular forms and Ramanujan tau | Ramanujan-style experimental evidence — variation 2 | Treat ramanujan-style experimental evidence as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M19M20M22 |
| 126 | 1930-1955 | Geometry of numbers and lattices | Lattice-point existence | Treat lattice-point existence as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M16M23M25 |
| 127 | 1930-1955 | Geometry of numbers and lattices | Convex-body bounds | Use convex-body bounds to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M23M25M31 |
| 128 | 1930-1955 | Geometry of numbers and lattices | Minkowski method in Diophantine problems | Read minkowski method in diophantine problems as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M25M31M32 |
| 129 | 1930-1955 | Geometry of numbers and lattices | Successive minima as arithmetic data | Reconstruct successive minima as arithmetic data through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M31M32M13 |
| 130 | 1930-1955 | Geometry of numbers and lattices | Forms and lattice geometry | Treat forms and lattice geometry as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M32M13M16 |
| 131 | 1930-1955 | Geometry of numbers and lattices | Counting lattice solutions | Use counting lattice solutions to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M13M16M23 |
| 132 | 1930-1955 | Geometry of numbers and lattices | Bounds for representation problems | Read bounds for representation problems as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M16M23M25 |
| 133 | 1930-1955 | Geometry of numbers and lattices | Geometry of numbers survey cases | Reconstruct geometry of numbers survey cases through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M23M25M31 |
| 134 | 1930-1955 | Geometry of numbers and lattices | Reduction domains | Treat reduction domains as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M25M31M32 |
| 135 | 1930-1955 | Geometry of numbers and lattices | Small determinant arguments | Use small determinant arguments to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M31M32M13 |
| 136 | 1930-1955 | Geometry of numbers and lattices | Local lattice obstructions | Read local lattice obstructions as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M32M13M16 |
| 137 | 1930-1955 | Geometry of numbers and lattices | Packing intuition for number theory | Reconstruct packing intuition for number theory through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M13M16M23 |
| 138 | 1930-1955 | Geometry of numbers and lattices | Finite search after bounding | Treat finite search after bounding as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M16M23M25 |
| 139 | 1930-1955 | Geometry of numbers and lattices | Explicit constants in geometry-of-numbers arguments | Use explicit constants in geometry-of-numbers arguments to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M23M25M31 |
| 140 | 1930-1955 | Geometry of numbers and lattices | Parametric lattices | Read parametric lattices as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M25M31M32 |
| 141 | 1930-1955 | Geometry of numbers and lattices | Exceptional low-dimensional cases | Reconstruct exceptional low-dimensional cases through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M31M32M13 |
| 142 | 1930-1955 | Geometry of numbers and lattices | Forms as geometric objects | Treat forms as geometric objects as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M32M13M16 |
| 143 | 1930-1955 | Geometry of numbers and lattices | Inequalities as Diophantine tools | Use inequalities as diophantine tools to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M13M16M23 |
| 144 | 1930-1955 | Geometry of numbers and lattices | Arithmetic domains and reduction | Read arithmetic domains and reduction as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M16M23M25 |
| 145 | 1930-1955 | Geometry of numbers and lattices | Geometry-first proof reconstruction | Reconstruct geometry-first proof reconstruction through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M23M25M31 |
| 146 | 1930-1955 | Geometry of numbers and lattices | Lattice-point existence — variation 2 | Treat lattice-point existence as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M25M31M32 |
| 147 | 1930-1955 | Geometry of numbers and lattices | Convex-body bounds — variation 2 | Use convex-body bounds to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M31M32M13 |
| 148 | 1930-1955 | Geometry of numbers and lattices | Minkowski method in Diophantine problems — variation 2 | Read minkowski method in diophantine problems as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M32M13M16 |
| 149 | 1930-1955 | Geometry of numbers and lattices | Successive minima as arithmetic data — variation 2 | Reconstruct successive minima as arithmetic data through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M13M16M23 |
| 150 | 1930-1955 | Geometry of numbers and lattices | Forms and lattice geometry — variation 2 | Treat forms and lattice geometry as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M16M23M25 |
| 151 | 1922-1969 | Higher genus and conjectural finiteness | Finiteness of rational points for genus greater than one | Treat finiteness of rational points for genus greater than one as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M12M25M26 |
| 152 | 1922-1969 | Higher genus and conjectural finiteness | Curves of genus zero, one, and greater than one | Use curves of genus zero, one, and greater than one to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M25M26M29 |
| 153 | 1922-1969 | Higher genus and conjectural finiteness | Genus as arithmetic phase transition | Read genus as arithmetic phase transition as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M26M29M33 |
| 154 | 1922-1969 | Higher genus and conjectural finiteness | Conjecture from elliptic success | Reconstruct conjecture from elliptic success through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M29M33M03 |
| 155 | 1922-1969 | Higher genus and conjectural finiteness | Rational point finiteness problem | Treat rational point finiteness problem as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M33M03M12 |
| 156 | 1922-1969 | Higher genus and conjectural finiteness | From finite basis to finite set | Use from finite basis to finite set to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M03M12M25 |
| 157 | 1922-1969 | Higher genus and conjectural finiteness | Examples suggesting scarcity | Read examples suggesting scarcity as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M12M25M26 |
| 158 | 1922-1969 | Higher genus and conjectural finiteness | Birational classification by genus | Reconstruct birational classification by genus through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M25M26M29 |
| 159 | 1922-1969 | Higher genus and conjectural finiteness | Problem formulation that outlives method | Treat problem formulation that outlives method as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M26M29M33 |
| 160 | 1922-1969 | Higher genus and conjectural finiteness | Historical path to Faltings | Use historical path to faltings to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M29M33M03 |
| 161 | 1922-1969 | Higher genus and conjectural finiteness | Comparison of rational and integral finiteness | Read comparison of rational and integral finiteness as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M33M03M12 |
| 162 | 1922-1969 | Higher genus and conjectural finiteness | Obstruction versus scarcity | Reconstruct obstruction versus scarcity through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M03M12M25 |
| 163 | 1922-1969 | Higher genus and conjectural finiteness | Families of high-genus curves | Treat families of high-genus curves as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M12M25M26 |
| 164 | 1922-1969 | Higher genus and conjectural finiteness | Diophantine geometry before the name | Use diophantine geometry before the name to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M25M26M29 |
| 165 | 1922-1969 | Higher genus and conjectural finiteness | Explicit high-genus examples | Read explicit high-genus examples as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M26M29M33 |
| 166 | 1922-1969 | Higher genus and conjectural finiteness | Problem-book transmission of the conjecture | Reconstruct problem-book transmission of the conjecture through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M29M33M03 |
| 167 | 1922-1969 | Higher genus and conjectural finiteness | Rational points as structural invariant | Treat rational points as structural invariant as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M33M03M12 |
| 168 | 1922-1969 | Higher genus and conjectural finiteness | From special cases to global principle | Use from special cases to global principle to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M03M12M25 |
| 169 | 1922-1969 | Higher genus and conjectural finiteness | Finiteness of rational points for genus greater than one — variation 2 | Read finiteness of rational points for genus greater than one as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M12M25M26 |
| 170 | 1922-1969 | Higher genus and conjectural finiteness | Curves of genus zero, one, and greater than one — variation 2 | Reconstruct curves of genus zero, one, and greater than one through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M25M26M29 |
| 171 | 1922-1969 | Higher genus and conjectural finiteness | Genus as arithmetic phase transition — variation 2 | Treat genus as arithmetic phase transition as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M26M29M33 |
| 172 | 1922-1969 | Higher genus and conjectural finiteness | Conjecture from elliptic success — variation 2 | Use conjecture from elliptic success to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M29M33M03 |
| 173 | 1922-1969 | Higher genus and conjectural finiteness | Rational point finiteness problem — variation 2 | Read rational point finiteness problem as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M33M03M12 |
| 174 | 1922-1969 | Higher genus and conjectural finiteness | From finite basis to finite set — variation 2 | Reconstruct from finite basis to finite set through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M03M12M25 |
| 175 | 1922-1969 | Higher genus and conjectural finiteness | Examples suggesting scarcity — variation 2 | Treat examples suggesting scarcity as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M12M25M26 |
| 176 | 1969 | Academic Press Diophantine Equations | Congruence impossibility chapter | Treat congruence impossibility chapter as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M28M27M26 |
| 177 | 1969 | Academic Press Diophantine Equations | Sums of squares | Use sums of squares to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M27M26M01 |
| 178 | 1969 | Academic Press Diophantine Equations | Quartic equations with trivial solutions | Read quartic equations with trivial solutions as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M26M01M03 |
| 179 | 1969 | Academic Press Diophantine Equations | Linear equations and congruences | Reconstruct linear equations and congruences through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M01M03M13 |
| 180 | 1969 | Academic Press Diophantine Equations | Pell equation | Treat pell equation as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M03M13M28 |
| 181 | 1969 | Academic Press Diophantine Equations | Rational solutions from known ones | Use rational solutions from known ones to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M13M28M27 |
| 182 | 1969 | Academic Press Diophantine Equations | Cubic curves | Read cubic curves as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M28M27M26 |
| 183 | 1969 | Academic Press Diophantine Equations | Cubic surfaces | Reconstruct cubic surfaces through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M27M26M01 |
| 184 | 1969 | Academic Press Diophantine Equations | Quartic surfaces | Treat quartic surfaces as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M26M01M03 |
| 185 | 1969 | Academic Press Diophantine Equations | Cubic equations in three variables | Use cubic equations in three variables to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M01M03M13 |
| 186 | 1969 | Academic Press Diophantine Equations | Algebraic number theory applications | Read algebraic number theory applications as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M03M13M28 |
| 187 | 1969 | Academic Press Diophantine Equations | Finite basis theorem chapter | Reconstruct finite basis theorem chapter through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M13M28M27 |
| 188 | 1969 | Academic Press Diophantine Equations | Rational points by genus | Treat rational points by genus as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M28M27M26 |
| 189 | 1969 | Academic Press Diophantine Equations | Binary quadratic and quaternary forms | Use binary quadratic and quaternary forms to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M27M26M01 |
| 190 | 1969 | Academic Press Diophantine Equations | Forms in several variables | Read forms in several variables as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M26M01M03 |
| 191 | 1969 | Academic Press Diophantine Equations | Thue theorem | Reconstruct thue theorem through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M01M03M13 |
| 192 | 1969 | Academic Press Diophantine Equations | Local p-adic methods | Treat local p-adic methods as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M03M13M28 |
| 193 | 1969 | Academic Press Diophantine Equations | Binary cubic forms | Use binary cubic forms to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M13M28M27 |
| 194 | 1969 | Academic Press Diophantine Equations | Binary quartic forms | Read binary quartic forms as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M28M27M26 |
| 195 | 1969 | Academic Press Diophantine Equations | Mordell equation chapter | Reconstruct mordell equation chapter through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M27M26M01 |
| 196 | 1969 | Academic Press Diophantine Equations | Higher-degree equations | Treat higher-degree equations as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M26M01M03 |
| 197 | 1969 | Academic Press Diophantine Equations | Fermat's last theorem survey | Use fermat's last theorem survey to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M01M03M13 |
| 198 | 1969 | Academic Press Diophantine Equations | Miscellaneous results | Read miscellaneous results as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M03M13M28 |
| 199 | 1969 | Academic Press Diophantine Equations | Methodological index of examples | Reconstruct methodological index of examples through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M13M28M27 |
| 200 | 1969 | Academic Press Diophantine Equations | Congruence impossibility chapter — variation 2 | Treat congruence impossibility chapter as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M28M27M26 |
| 201 | 1930-1972 | Selected papers and exposition | Problem-centered exposition | Treat problem-centered exposition as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M27M28M29 |
| 202 | 1930-1972 | Selected papers and exposition | Discursive proof style | Use discursive proof style to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M28M29M31 |
| 203 | 1930-1972 | Selected papers and exposition | Examples before definitions | Read examples before definitions as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M29M31M33 |
| 204 | 1930-1972 | Selected papers and exposition | Historical comments as method | Reconstruct historical comments as method through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M31M33M26 |
| 205 | 1930-1972 | Selected papers and exposition | Review essays and priority notes | Treat review essays and priority notes as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M33M26M27 |
| 206 | 1930-1972 | Selected papers and exposition | Lecture-to-monograph transformation | Use lecture-to-monograph transformation to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M26M27M28 |
| 207 | 1930-1972 | Selected papers and exposition | Arithmetic taste as pedagogy | Read arithmetic taste as pedagogy as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M27M28M29 |
| 208 | 1930-1972 | Selected papers and exposition | Explicit calculation in exposition | Reconstruct explicit calculation in exposition through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M28M29M31 |
| 209 | 1930-1972 | Selected papers and exposition | Notation kept close to examples | Treat notation kept close to examples as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M29M31M33 |
| 210 | 1930-1972 | Selected papers and exposition | Bibliographic cross-references | Use bibliographic cross-references to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M31M33M26 |
| 211 | 1930-1972 | Selected papers and exposition | Problem list as theorem map | Read problem list as theorem map as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M33M26M27 |
| 212 | 1930-1972 | Selected papers and exposition | Mature view of Diophantine equations | Reconstruct mature view of diophantine equations through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M26M27M28 |
| 213 | 1930-1972 | Selected papers and exposition | Reader-guided reconstruction | Treat reader-guided reconstruction as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M27M28M29 |
| 214 | 1930-1972 | Selected papers and exposition | Comparison of proof routes | Use comparison of proof routes to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M28M29M31 |
| 215 | 1930-1972 | Selected papers and exposition | Mathematical memory through examples | Read mathematical memory through examples as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M29M31M33 |
| 216 | 1930-1972 | Selected papers and exposition | Open problems as closing structure | Reconstruct open problems as closing structure through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M31M33M26 |
| 217 | 1930-1972 | Selected papers and exposition | Problem-centered exposition — variation 2 | Treat problem-centered exposition as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M33M26M27 |
| 218 | 1930-1972 | Selected papers and exposition | Discursive proof style — variation 2 | Use discursive proof style to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M26M27M28 |
| 219 | 1930-1972 | Selected papers and exposition | Examples before definitions — variation 2 | Read examples before definitions as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M27M28M29 |
| 220 | 1930-1972 | Selected papers and exposition | Historical comments as method — variation 2 | Reconstruct historical comments as method through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M28M29M31 |
| 221 | 1930-1972 | Selected papers and exposition | Review essays and priority notes — variation 2 | Treat review essays and priority notes as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M29M31M33 |
| 222 | 1930-1972 | Selected papers and exposition | Lecture-to-monograph transformation — variation 2 | Use lecture-to-monograph transformation to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M31M33M26 |
| 223 | 1930-1972 | Selected papers and exposition | Arithmetic taste as pedagogy — variation 2 | Read arithmetic taste as pedagogy as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M33M26M27 |
| 224 | 1930-1972 | Selected papers and exposition | Explicit calculation in exposition — variation 2 | Reconstruct explicit calculation in exposition through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M26M27M28 |
| 225 | 1930-1972 | Selected papers and exposition | Notation kept close to examples — variation 2 | Treat notation kept close to examples as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M27M28M29 |
| 226 | 1922-1945 | Manchester and research ecology | Recruiting visitors around number theory | Treat recruiting visitors around number theory as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M30M26M29 |
| 227 | 1922-1945 | Manchester and research ecology | Problem ecology at Manchester | Use problem ecology at manchester to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M26M29M25 |
| 228 | 1922-1945 | Manchester and research ecology | Young mathematicians and concrete questions | Read young mathematicians and concrete questions as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M29M25M33 |
| 229 | 1922-1945 | Manchester and research ecology | Department-building as research method | Reconstruct department-building as research method through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M25M33M31 |
| 230 | 1922-1945 | Manchester and research ecology | Seminar problems and arithmetic culture | Treat seminar problems and arithmetic culture as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M33M31M30 |
| 231 | 1922-1945 | Manchester and research ecology | Davenport-era problem circulation | Use davenport-era problem circulation to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M31M30M26 |
| 232 | 1922-1945 | Manchester and research ecology | Mahler-style interaction | Read mahler-style interaction as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M30M26M29 |
| 233 | 1922-1945 | Manchester and research ecology | Erdos visit as problem exchange | Reconstruct erdos visit as problem exchange through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M26M29M25 |
| 234 | 1922-1945 | Manchester and research ecology | Segre and algebraic geometry contact | Treat segre and algebraic geometry contact as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M29M25M33 |
| 235 | 1922-1945 | Manchester and research ecology | Transmission of Diophantine problems | Use transmission of diophantine problems to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M25M33M31 |
| 236 | 1922-1945 | Manchester and research ecology | Concrete arithmetic as shared language | Read concrete arithmetic as shared language as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M33M31M30 |
| 237 | 1922-1945 | Manchester and research ecology | Problem families across visitors | Reconstruct problem families across visitors through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M31M30M26 |
| 238 | 1922-1945 | Manchester and research ecology | Institutional memory through examples | Treat institutional memory through examples as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M30M26M29 |
| 239 | 1922-1945 | Manchester and research ecology | Research taste formation | Use research taste formation to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M26M29M25 |
| 240 | 1922-1945 | Manchester and research ecology | Recruiting visitors around number theory — variation 2 | Read recruiting visitors around number theory as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M29M25M33 |
| 241 | 1922-1945 | Manchester and research ecology | Problem ecology at Manchester — variation 2 | Reconstruct problem ecology at manchester through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M25M33M31 |
| 242 | 1922-1945 | Manchester and research ecology | Young mathematicians and concrete questions — variation 2 | Treat young mathematicians and concrete questions as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M33M31M30 |
| 243 | 1922-1945 | Manchester and research ecology | Department-building as research method — variation 2 | Use department-building as research method to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M31M30M26 |
| 244 | 1922-1945 | Manchester and research ecology | Seminar problems and arithmetic culture — variation 2 | Read seminar problems and arithmetic culture as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M30M26M29 |
| 245 | 1922-1945 | Manchester and research ecology | Davenport-era problem circulation — variation 2 | Reconstruct davenport-era problem circulation through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M26M29M25 |
| 246 | 1922-1945 | Manchester and research ecology | Mahler-style interaction — variation 2 | Treat mahler-style interaction as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M29M25M33 |
| 247 | 1922-1945 | Manchester and research ecology | Erdos visit as problem exchange — variation 2 | Use erdos visit as problem exchange to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M25M33M31 |
| 248 | 1922-1945 | Manchester and research ecology | Segre and algebraic geometry contact — variation 2 | Read segre and algebraic geometry contact as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M33M31M30 |
| 249 | 1922-1945 | Manchester and research ecology | Transmission of Diophantine problems — variation 2 | Reconstruct transmission of diophantine problems through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M31M30M26 |
| 250 | 1922-1945 | Manchester and research ecology | Concrete arithmetic as shared language — variation 2 | Treat concrete arithmetic as shared language as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M30M26M29 |
| 251 | 1920-1965 | Inequalities, elementary number theory, miscellany | Erdos-Mordell inequality style geometry | Treat erdos-mordell inequality style geometry as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M02M23M31 |
| 252 | 1920-1965 | Inequalities, elementary number theory, miscellany | Elementary inequalities with arithmetic flavor | Use elementary inequalities with arithmetic flavor to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M23M31M32 |
| 253 | 1920-1965 | Inequalities, elementary number theory, miscellany | Residue classes in elementary problems | Read residue classes in elementary problems as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M31M32M27 |
| 254 | 1920-1965 | Inequalities, elementary number theory, miscellany | Special polynomial representations | Reconstruct special polynomial representations through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M32M27M17 |
| 255 | 1920-1965 | Inequalities, elementary number theory, miscellany | Small counterexamples | Treat small counterexamples as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M27M17M02 |
| 256 | 1920-1965 | Inequalities, elementary number theory, miscellany | Impossible equations by congruence | Use impossible equations by congruence to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M17M02M23 |
| 257 | 1920-1965 | Inequalities, elementary number theory, miscellany | Finite exceptional cases | Read finite exceptional cases as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M02M23M31 |
| 258 | 1920-1965 | Inequalities, elementary number theory, miscellany | Concrete estimates | Reconstruct concrete estimates through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M23M31M32 |
| 259 | 1920-1965 | Inequalities, elementary number theory, miscellany | Number-theoretic identities | Treat number-theoretic identities as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M31M32M27 |
| 260 | 1920-1965 | Inequalities, elementary number theory, miscellany | Elementary proof variants | Use elementary proof variants to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M32M27M17 |
| 261 | 1920-1965 | Inequalities, elementary number theory, miscellany | Comparison of solution routes | Read comparison of solution routes as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M27M17M02 |
| 262 | 1920-1965 | Inequalities, elementary number theory, miscellany | Miscellaneous Diophantine examples | Reconstruct miscellaneous diophantine examples through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M17M02M23 |
| 263 | 1920-1965 | Inequalities, elementary number theory, miscellany | Transforming olympiad-like problems | Treat transforming olympiad-like problems as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M02M23M31 |
| 264 | 1920-1965 | Inequalities, elementary number theory, miscellany | From inequality to invariant | Use from inequality to invariant to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M23M31M32 |
| 265 | 1920-1965 | Inequalities, elementary number theory, miscellany | Erdos-Mordell inequality style geometry — variation 2 | Read erdos-mordell inequality style geometry as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M31M32M27 |
| 266 | 1920-1965 | Inequalities, elementary number theory, miscellany | Elementary inequalities with arithmetic flavor — variation 2 | Reconstruct elementary inequalities with arithmetic flavor through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M32M27M17 |
| 267 | 1920-1965 | Inequalities, elementary number theory, miscellany | Residue classes in elementary problems — variation 2 | Treat residue classes in elementary problems as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M27M17M02 |
| 268 | 1920-1965 | Inequalities, elementary number theory, miscellany | Special polynomial representations — variation 2 | Use special polynomial representations to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M17M02M23 |
| 269 | 1920-1965 | Inequalities, elementary number theory, miscellany | Small counterexamples — variation 2 | Read small counterexamples as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M02M23M31 |
| 270 | 1920-1965 | Inequalities, elementary number theory, miscellany | Impossible equations by congruence — variation 2 | Reconstruct impossible equations by congruence through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M23M31M32 |
| 271 | 1920-1965 | Inequalities, elementary number theory, miscellany | Finite exceptional cases — variation 2 | Treat finite exceptional cases as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M31M32M27 |
| 272 | 1920-1965 | Inequalities, elementary number theory, miscellany | Concrete estimates — variation 2 | Use concrete estimates to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M32M27M17 |
| 273 | 1920-1965 | Inequalities, elementary number theory, miscellany | Number-theoretic identities — variation 2 | Read number-theoretic identities as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M27M17M02 |
| 274 | 1920-1965 | Inequalities, elementary number theory, miscellany | Elementary proof variants — variation 2 | Reconstruct elementary proof variants through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M17M02M23 |
| 275 | 1920-1965 | Inequalities, elementary number theory, miscellany | Comparison of solution routes — variation 2 | Treat comparison of solution routes as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M02M23M31 |
| 276 | 1972-present | Legacy and modern reinterpretation | Mordell-Weil theorem in modern language | Treat mordell-weil theorem in modern language as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M08M09M10 |
| 277 | 1972-present | Legacy and modern reinterpretation | Mordell curves in computational arithmetic | Use mordell curves in computational arithmetic to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M09M10M12 |
| 278 | 1972-present | Legacy and modern reinterpretation | Heights after Tate and Neron | Read heights after tate and neron as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M10M12M25 |
| 279 | 1972-present | Legacy and modern reinterpretation | Faltings theorem as Mordell conjecture resolution | Reconstruct faltings theorem as mordell conjecture resolution through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M12M25M33 |
| 280 | 1972-present | Legacy and modern reinterpretation | Elliptic curve rank as finite-basis problem | Treat elliptic curve rank as finite-basis problem as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M25M33M08 |
| 281 | 1972-present | Legacy and modern reinterpretation | Mordell equation databases | Use mordell equation databases to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M33M08M09 |
| 282 | 1972-present | Legacy and modern reinterpretation | Selmer groups and descent after Mordell | Read selmer groups and descent after mordell as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M08M09M10 |
| 283 | 1972-present | Legacy and modern reinterpretation | Rational points as Diophantine geometry | Reconstruct rational points as diophantine geometry through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M09M10M12 |
| 284 | 1972-present | Legacy and modern reinterpretation | Modern p-adic methods revisiting old problems | Treat modern p-adic methods revisiting old problems as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M10M12M25 |
| 285 | 1972-present | Legacy and modern reinterpretation | Mordell-Lang perspective | Use mordell-lang perspective to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M12M25M33 |
| 286 | 1972-present | Legacy and modern reinterpretation | Arithmetic geometry reconstruction | Read arithmetic geometry reconstruction as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M25M33M08 |
| 287 | 1972-present | Legacy and modern reinterpretation | Computational search for generators | Reconstruct computational search for generators through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M33M08M09 |
| 288 | 1972-present | Legacy and modern reinterpretation | Modern textbooks on Mordell-Weil | Treat modern textbooks on mordell-weil as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M08M09M10 |
| 289 | 1972-present | Legacy and modern reinterpretation | Historical rereading of finite basis | Use historical rereading of finite basis to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M09M10M12 |
| 290 | 1972-present | Legacy and modern reinterpretation | Curves by genus in modern form | Read curves by genus in modern form as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M10M12M25 |
| 291 | 1972-present | Legacy and modern reinterpretation | Mordell's problem style in contemporary research | Reconstruct mordell's problem style in contemporary research through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M12M25M33 |
| 292 | 1972-present | Legacy and modern reinterpretation | Mordell-Weil theorem in modern language — variation 2 | Treat mordell-weil theorem in modern language as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M25M33M08 |
| 293 | 1972-present | Legacy and modern reinterpretation | Mordell curves in computational arithmetic — variation 2 | Use mordell curves in computational arithmetic to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M33M08M09 |
| 294 | 1972-present | Legacy and modern reinterpretation | Heights after Tate and Neron — variation 2 | Read heights after tate and neron as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M08M09M10 |
| 295 | 1972-present | Legacy and modern reinterpretation | Faltings theorem as Mordell conjecture resolution — variation 2 | Reconstruct faltings theorem as mordell conjecture resolution through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M09M10M12 |
| 296 | 1972-present | Legacy and modern reinterpretation | Elliptic curve rank as finite-basis problem — variation 2 | Treat elliptic curve rank as finite-basis problem as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M10M12M25 |
| 297 | 1972-present | Legacy and modern reinterpretation | Mordell equation databases — variation 2 | Use mordell equation databases to pass from examples to arithmetic structure: record congruences, transformations, height behavior, and exceptional cases before abstracting. | M12M25M33 |
| 298 | 1972-present | Legacy and modern reinterpretation | Selmer groups and descent after Mordell — variation 2 | Read selmer groups and descent after mordell as a Mordell-style case study: begin with hand-computable instances, identify the invariant, and promote the observed pattern into a theorem or problem family. | M25M33M08 |
| 299 | 1972-present | Legacy and modern reinterpretation | Rational points as Diophantine geometry — variation 2 | Reconstruct rational points as diophantine geometry through the triad of local evidence, global descent, and explicit representation, keeping rational and integral questions separate. | M33M08M09 |
| 300 | 1972-present | Legacy and modern reinterpretation | Modern p-adic methods revisiting old problems — variation 2 | Treat modern p-adic methods revisiting old problems as a concrete Diophantine problem: normalize the equation, expose the relevant obstruction, and decide whether construction, descent, or finiteness is the governing move. | M08M09M10 |
04
Source Spine
MacTutor biography
Biographical chronology and research context: Philadelphia birth, Cambridge education, Manchester/Cambridge career, finite basis theorem, Mordell conjecture, and later lecture-style book.
Source linkBiographical Memoirs / Cassels
Cassels's memoir is the classical biographical account of Mordell's mathematical personality, work, and influence.
Source linkDiophantine Equations
Academic Press monograph organizing Mordell's mature view: congruences, Pell equations, cubic curves, finite basis theorem, genus, forms, p-adic methods, Mordell curves, and higher-degree equations.
Source linkPrinceton catalog contents
Library contents list for Mordell's Diophantine Equations, useful for reconstructing the chapter-level source spine.
Source linkMordell-Weil theorem references
Modern theorem statement: rational points on an elliptic curve over Q are finitely generated; Weil extended the theorem to abelian varieties over number fields.
Source linkYouthful writings on y^2-k=x^3
Historical analysis of Mordell's 1914 work on the equation y^2-k=x^3 via reciprocity, ideals, and binary cubic forms.
Source link05
Worked Demonstrations
Finite basis theorem as workflow
\[E(\mathbb Q)\cong \mathbb Z^r\oplus E(\mathbb Q)_{\rm tors}\]
- Choose a cubic with a rational point and impose the chord-tangent composition law.
- Separate torsion phenomena from points whose height grows under repeated addition.
- Use descent to show that rational points reduce to finitely many height-bounded representatives.
- Promote the computation into the finite-basis theorem for rational points on a genus-one cubic.
Mordell curve laboratory
\[y^2=x^3+k\]
- Fix k and test congruence obstructions modulo small primes.
- Factor in auxiliary quadratic or cubic rings when the equation demands it.
- Use descent or form reduction to bound the possible primitive solutions.
- Compare many k values to decide whether the behavior is accidental or structural.
From theorem to conjecture
\[g(X)>1\Rightarrow \#X(\mathbb Q)<\infty\]
- Observe that genus-one curves have a finitely generated group rather than arbitrary chaos.
- Notice that higher genus removes the group-law mechanism while preserving the rational-point problem.
- Formulate a finiteness claim strong enough to survive the loss of parametrization.
- Leave a problem whose eventual proof reshapes Diophantine geometry.
Forms as translation devices
\[F(x,y)=m\quad\leftrightarrow\quad \text{ideal/class data}\]
- Transform the original equation into a binary form or norm equation.
- Classify the form by discriminant and equivalence transformations.
- Use ideals, composition, or reduction theory to control the solution set.
- Translate the result back into rational or integral points on the original curve.
Ramanujan tau multiplicativity
\[\tau(mn)=\tau(m)\tau(n)\quad((m,n)=1)\]
- Begin with a q-series whose coefficients appear arithmetically structured.
- Test the coefficient pattern against products of relatively prime indices.
- Prove that multiplicativity follows from the modular or product structure behind the series.
- Treat the coefficient sequence as an arithmetic object, not a table.
Monograph as method atlas
\[\text{chapter}\simeq\text{problem family}+\text{method inventory}\]
- Group equations by method rather than by superficial degree alone.
- For each class, record impossibility tests, transformations, examples, and known theorems.
- Preserve exceptions, historical notes, and open directions as part of the mathematical data.
- Make the reader inherit a research program, not just a collection of solved exercises.