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computational-explorations

Computational explorations in number theory, combinatorics, and beyond. Coprime Ramsey numbers, Schur extensions, survival analysis of mathematical problems, and more.

Started 2026 Python

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Computational Explorations in Number Theory, Combinatorics, and Beyond

A curiosity-driven research project exploring mathematical structure through computation, spanning Erdős problems, Ramsey theory, additive combinatorics, and cross-domain pattern discovery.

Headline Results

Discovery Value Status
Coprime Ramsey R_cop(3) 11 Exact (incremental extension)
Coprime Ramsey R_cop(4) 59 SAT lower bound + extension upper bound
Multi-color R_cop(3; 3) 53 Exact (SAT)
25+ coprime Ramsey variants Full table Exact (SAT)
S(G,k) order-invariance Holds k≤2, fails k=3 Verified order ≤ 20
Coprime graph perfectness χ = ω = 1+π(n) Verified n ≤ 30 (prior art exists)
Coprime Ramsey primality All 4 known values prime Conjecture (NPG-30)

Repository Structure

src/                    # 56+ Python modules — computational experiments
tests/                  # 57+ test files, ~3000 tests
lean/                   # Lean 4 formalizations (2 complete, 0 sorry)
paper/                  # LaTeX papers for DOI minting
docs/                   # Research findings, provenance, OEIS candidates
discoveries/            # Organized discovery summaries
  coprime-ramsey/       # The coprime Ramsey theory we built
  schur-extensions/     # Schur number extensions (S(G,k), DS, MS)
  meta-analysis/        # Mathematical sociology, AI acceleration
  cross-domain/         # Connections to CS, stats, physics
prior-art/              # Literature references and provenance checks
proofs/                 # Informal proof documents
data/                   # Erdős problem database (from teorth/erdosproblems)

Key Documents

Conjectures

ID Statement Evidence
NPG-27 S(G,k) depends only on |G| for abelian groups, k ≤ 2 All groups order ≤ 20
NPG-28 DS(2,α) has exact phase transition thresholds Full phase diagram computed
NPG-29 R_cop(4) = 59 SAT + extension proof
NPG-30 Coprime clique Ramsey numbers are always prime 4/4: {2, 11, 53, 59}
NPG-31 Coprime graph G(n) is perfect SAT n ≤ 30; prior art for groups

Methodology

  • Python with numpy, scipy, sklearn, pysat (SAT solving)
  • Lean 4 with Mathlib for formal proofs
  • SAT solvers: Glucose4, CaDiCaL195 via python-sat
  • AI assistance: Claude (Anthropic) — all results independently verified

Provenance

  • Erdős database: teorth/erdosproblems
  • Coprime graph perfectness: Prior art — Syarifudin & Wardhana (groups case)
  • Product Schur triples: Related work — Mattos et al. (SIAM J. Discrete Math, 2024)
  • Coprime Ramsey numbers R_cop(k): Novel (no prior art found, March 2026)
  • Full provenance report: docs/provenance_and_verification.md

Author

Alex Towell — M.S. Mathematics, M.S. Computer Science, Ph.D. student (CS), SIUE atowell@siue.edu · GitHub

License

Code: MIT. Papers: CC-BY-4.0.

DOIs

Paper DOI
Coprime Ramsey Numbers 10.5281/zenodo.19058647
Schur Number Extensions 10.5281/zenodo.19058653
Survival Analysis of Erdős Problems 10.5281/zenodo.19058655

Discussion