Surveying Competition Problems: Erdős–Straus as Most Accessible Unsolved Conjecture — E8 Intelligence Research

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 \(n \geq 2\), \(\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\) 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 \(n \equiv 1 \pmod{4}\) and \

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com