active library

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

Provenance & Verification — Prior art checks, novelty assessments
DOI Candidates — Artifacts ready for permanent citation
Session Findings — Detailed discovery log
Novel Conjectures — NPG-27 through NPG-31
OEIS Candidates — Sequences for submission

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