name math-techniques description Mathematical proof techniques library for working on Erdős problems. Use when attempting to solve combinatorics, number theory, graph theory, or geometry problems. Includes techniques from successful AI solutions. Mathematical Techniques Library A curated collection of proof techniques relevant to Erdős problems, organized by problem domain. Quick Reference Domain Common Techniques Number Theory Pell equations, quadratic forms, modular arithmetic Combinatorics Pigeonhole, probabilistic method, counting arguments Graph Theory Ramsey theory, chromatic bounds, incidence geometry Geometry Lattice constructions, distance sets, polynomial methods Technique Files For detailed technique descriptions, see: NUMBER_THEORY.md

Prime sequences, Diophantine equations COMBINATORICS.md
Counting, Ramsey, extremal GRAPH_THEORY.md
Coloring, distances, matchings GEOMETRY.md
Point sets, distances, lattices LITERATURE.md
Key papers and reductions Problem-Solving Workflow Phase 1: Problem Analysis Parse the LaTeX statement carefully Identify the problem domain (tags) Check if formalized in Lean (lean_url field) Read original Erdős paper references Phase 2: Literature Check Search for problem number in academic databases Check Terence Tao's wiki for AI progress Look for related OEIS sequences Review comments on erdosproblems.com Phase 3: Technique Selection Based on problem type: Existence proofs : Probabilistic method, constructions Bounds : Incidence geometry, polynomial methods Sequences : Pell equations, growth rate analysis Graphs : Ramsey theory, chromatic number bounds Phase 4: Solution Attempt Try simplest applicable technique first Look for reductions to known results Check for counterexamples if conjecture Verify small cases computationally Phase 5: Verification Check solution addresses intended interpretation Search literature for existing solutions Verify with formal tools if possible Have human expert review Success Patterns from AI Solutions Erdős-652 (Distinct Distances) Technique : Reduction to Pach-Sharir incidence bounds Pattern : Connect discrete geometry to incidence theory Erdős-1051 (Irrationality of Series) Technique : Mahler's criterion, growth rate analysis Pattern : Establish contradiction via asymptotic bounds Erdős-654 (Four Points No Circle) Technique : Axis-aligned construction with prime powers Pattern : Use number-theoretic constraints for geometric problems Erdős-935 (Powerful Parts) Technique : Pell equation solutions + Dirichlet's theorem Pattern : Construct specific sequences via Diophantine equations Erdős-397 (Central Binomial Products) Technique : Explicit parametric family construction Pattern : Find algebraic identity yielding infinite solutions Warning Signs Problem seems "too easy" → likely solved or misinterpreted Solution doesn't use domain-specific tools → check interpretation Can't find any related literature → problem may be stated incorrectly Proof works for all cases trivially → definitional ambiguity likely

Prime sequences, Diophantine equations COMBINATORICS.md
Counting, Ramsey, extremal GRAPH_THEORY.md
Coloring, distances, matchings GEOMETRY.md
Point sets, distances, lattices LITERATURE.md
Key papers and reductions Problem-Solving Workflow Phase 1: Problem Analysis Parse the LaTeX statement carefully Identify the problem domain (tags) Check if formalized in Lean (lean_url field) Read original Erdős paper references Phase 2: Literature Check Search for problem number in academic databases Check Terence Tao's wiki for AI progress Look for related OEIS sequences Review comments on erdosproblems.com Phase 3: Technique Selection Based on problem type: Existence proofs : Probabilistic method, constructions Bounds : Incidence geometry, polynomial methods Sequences : Pell equations, growth rate analysis Graphs : Ramsey theory, chromatic number bounds Phase 4: Solution Attempt Try simplest applicable technique first Look for reductions to known results Check for counterexamples if conjecture Verify small cases computationally Phase 5: Verification Check solution addresses intended interpretation Search literature for existing solutions Verify with formal tools if possible Have human expert review Success Patterns from AI Solutions Erdős-652 (Distinct Distances) Technique : Reduction to Pach-Sharir incidence bounds Pattern : Connect discrete geometry to incidence theory Erdős-1051 (Irrationality of Series) Technique : Mahler's criterion, growth rate analysis Pattern : Establish contradiction via asymptotic bounds Erdős-654 (Four Points No Circle) Technique : Axis-aligned construction with prime powers Pattern : Use number-theoretic constraints for geometric problems Erdős-935 (Powerful Parts) Technique : Pell equation solutions + Dirichlet's theorem Pattern : Construct specific sequences via Diophantine equations Erdős-397 (Central Binomial Products) Technique : Explicit parametric family construction Pattern : Find algebraic identity yielding infinite solutions Warning Signs Problem seems "too easy" → likely solved or misinterpreted Solution doesn't use domain-specific tools → check interpretation Can't find any related literature → problem may be stated incorrectly Proof works for all cases trivially → definitional ambiguity likely

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