Surveying Competition Problems: Erdős–Straus as Most Accessible Unsolved Conjecture — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Survey of open competition-style problems reveals no new solved mathematics, but highlights the Erdős–Straus conjecture as the most structurally accessible unsolved problem, alongside Putnam's hardest problem as a solved exemplar of elegant technique. | MATH: Erdős–Straus conjecture: For every integer n ≥ 2, 4/n = 1/x + 1/y + 1/z has positive integer solutions x,y,z. No counterexample known; unproven for all n. Putnam hardest problem (1985 B6?): Typically involves sums of reciprocals or combinatorial identities — exact equation not extractable from metadata. | CONNECTION: The Erdős–Straus equation is a **unit fraction decomposition** — its structure mirrors **Egyptian fraction** expansions, which relate to **harmonic series** and **modular arithmetic**. The denominator n modulo small primes (2,3,5,7) yields residue classes where solutions are known; the obstruction lies in residue classes like n ≡ 1 pmod4 and \
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- Lack of Accessible 2025/2026 IMO Problems in Search Results — E8 Intelligence Research
- Competition Problems vs. Genuine Discovery: Hilbert Geometry as the Sole Novel Result — E8 Intelligence Research
- Unsolved Math Problems and Competition Math Video Compilation — E8 Intelligence Research
- Putnam Problems All Solved; Only Novelty Is Proposed Physics Competition — E8 Intelligence Research
- Cayley Graph Proof of the Collatz Conjecture via Decidable an+b Problems — E8 Intelligence Research
BibTeX
@misc{caldin2026surveying,
title = {{Surveying Competition Problems: Erdős–Straus as Most Accessible Unsolved Conjecture — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23553557},
url = {https://zenodo.org/records/23553557},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Surveying Competition Problems: Erdős–Straus as Most Accessible Unsolved Conjecture — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/22
PB - Zenodo
DO - 10.5281/zenodo.23553557
UR - https://zenodo.org/records/23553557
AB - FINDING: Survey of open competition-style problems reveals no new solved mathematics, but highlights the Erdős–Straus conjecture as the most structurally accessible unsolved problem, alongside Putnam's hardest problem as a solved exemplar of elegant technique. | MATH: Erdős–Straus conjecture: For every integer n ≥ 2, 4/n = 1/x + 1/y + 1/z has positive integer solutions x,y,z. No counterexample known; unproven for all n. Putnam hardest problem (1985 B6?): Typically involves sums of reciprocals or combinatorial identities — exact equation not extractable from metadata. | CONNECTION: The Erdős–Straus equation is a **unit fraction decomposition** — its structure mirrors **Egyptian fraction** expansions, which relate to **harmonic series** and **modular arithmetic**. The denominator n modulo small primes (2,3,5,7) yields residue classes where solutions are known; the obstruction lies in residue classes like n ≡ 1 pmod4 and \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -