slug: oeis-a115039-prime-square-perfect-power-representation bibkey: hilliard2006a115039 doi: null url: https://oeis.org/A115039 triage: theorem motivation_gids:
- D5/S3/Arith/PrimeSquarePerfectPowerRepresentation
Prime square perfect-power representation
Problem
OEIS A115039 and A115038 state the same conjecture:
Conjecture: There will always be an x,y,n such that x^2 + p = y^n for all primes p.
The quantified reading is: for every prime natural number p, there are
natural numbers x, y, and n with 2 <= n and
x^2 + p = y^n. The lower bound is the standard perfect-power condition;
without it, exponent one makes the equation immediate.
Motivation
The assertion asks for one nontrivial perfect-power representation for every prime, including the even prime.
Gap
At the 2026-09-17 reading, both entries still label the assertion a conjecture. Repository, pinned Mathlib, Loogle, GitHub Lean-code, and scoped literature searches found no theorem with the complete conclusion.
The search was not an exhaustive global priority investigation.
Route
For an odd prime, write p = 2t + 1. Then
t^2 + p = t^2 + 2t + 1 = (t + 1)^2.
For the prime two, 5^2 + 2 = 3^3. These choices have positive bases and
exponents at least two.
Falsifier
A prime for which no natural witnesses satisfy the quantified equation would refute the assertion. Finite searches can detect a bad construction but cannot establish the universal statement.
Evidence
D5/S3/Arith/PrimeSquarePerfectPowerRepresentation.result proves the full
quantified statement by the parity split above.
Triage
theorem; resolution proved.
ASSUMED-UNVERIFIED
No exhaustive global priority claim is made beyond the searched surfaces.