The Diophantine Equation 2^p - p = k^m: Closed Exponent Classes, a Uniform Exponent Bound, and Effective Finiteness

Abstract

We study the equation 2^p − p = k^m in primes p and integers k, m ≥ 2. Exactly two solutions are known, (p,k,m) = (5,3,3) and (7,11,2), and an exhaustive computation reported here shows there are no others with p ≤ 10^6. Results. (A) The only solution with m even is (7,11,2); the proof combines Beukers' explicit bounds for the generalized Ramanujan–Nagell equation x² − D = 2^n with a mod-8 filter and is checkable by hand (p ≤ 27). (B) The only solution with 3 | m is (5,3,3), via Bauer–Bennett's theorem on Ramanujan–Nagell cubics (p ≤ 11). (C) Every solution with p ≥ 10 satisfies m < 52,414 uniformly in p: we repair a gap in an earlier derivation from the Laurent–Mignotte–Nesterenko two-logarithm bounds, whose parameter estimate had silently restricted the range to p ≤ 1.17×10^9. (D) Combining (A)–(C) with effective bounds for Thue equations (Bugeaud–Győry 1996), the equation has only finitely many solutions, all effectively bounded. The effective bound is astronomically large, so the conjecture that no third solution exists remains open. Contents. The deposit contains the research note (PDF + LaTeX source), a README, and a reproducibility package: all verification scripts (Python 3, standard library only, exact integer arithmetic), the itemized proof-status ledger, detailed write-ups with citation caveats and falsifiability notes, and computation logs for the exhaustive search up to p = 10^6. Transparency / AI disclosure. The mathematical content was developed and adversarially cross-verified with substantial assistance from an AI system (Claude, Anthropic), directed and reviewed by the author, and has not yet been reviewed by a human expert. Section 9 of the note records precisely which cited results were read at their published source and which were verified only via secondary literature. The material is published so that anyone who wishes to verify, correct, or extend it has everything needed to do so.