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October 15, 2025Open Access

On the Lebesgue-Nagell equation x²-2 = yᵖ

Key Points

The findings confirm that the only solutions for the Lebesgue-Nagell equation are trivial for p ≤ 13.
For primes p > 911, the conjecture is proven, showing nontrivial solutions would require y to be exceedingly large.
Lower bounds for linear forms in logarithms were crucial in establishing results for larger primes.
The results also revisit previous claims, suggesting paths for future research into integer solution families.

Abstract

We investigate the Lebesgue--Nagell equation align* x²-2=yᵖ align* in integers x, y, p with p 3 an odd prime. A longstanding folklore conjecture asserts that the only solutions are the ``trivial'' ones with y=-1. We confirm the conjecture unconditionally for p 13, and prove the conjecture holds for p>911 through a careful application of lower bounds for linear forms in two logarithms. We also show that any ``nontrivial'' solution must satisfy y > 10^1000. In addition, we establish auxiliary results that may support future progress on the problem, and we revisit some prior claims in the literature.

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