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Interpolation determinants and the Lebesgue--Nagell equation a2−D=bp
Abstract
We prove the long-standing conjecture that, for every odd prime p, the only integral solutions of a2−2=bp are (a,b)=(± 1, −1). We also completely solve the analogous equations y2−D=xp for D=3 and D=5 and describe a general approach for other positive squarefree values of D not ≡ 1 pmod 8. The proof refines the interpolation determinant method for linear forms in two logarithms in special cases, introducing new ideas for both the arithmetic lower bounds and the analytic upper bounds.
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BibTeX
@misc{lombardo2026interpolation,
title = {{Interpolation determinants and the Lebesgue--Nagell equation $a^2-D=b^p$}},
author = {Davide Lombardo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23899},
eprint = {2609.23899},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23899},
}
RIS
TY - UNPB
TI - Interpolation determinants and the Lebesgue--Nagell equation a^2−D=b^p
AU - Davide Lombardo
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23899
UR - https://arxiv.org/abs/2609.23899
AB - We prove the long-standing conjecture that, for every odd prime p, the only integral solutions of a^2−2=b^p are (a,b)=(± 1, −1). We also completely solve the analogous equations y^2−D=x^p for D=3 and D=5 and describe a general approach for other positive squarefree values of D not ≡ 1 pmod 8. The proof refines the interpolation determinant method for linear forms in two logarithms in special cases, introducing new ideas for both the arithmetic lower bounds and the analytic upper bounds.
ER -