AI 中文总结
本文证明奇素数$p$下$a^2-2=b^p$仅有平凡解，并解决$D=3,5$的类似方程，提出一般方法，改进插值行列式技术。
AI 中文摘要
我们证明了长期存在的猜想：对于每个奇素数$p$，方程$a^2-2=b^p$的唯一整数解是$(a,b)=(\pm 1, -1)$。我们还完全解决了$D=3$和$D=5$时的类似方程$y^2-D=x^p$，并描述了其他$D \not\equiv 1 \pmod 8$的正无平方因子值的一般方法。该证明在特殊情形下改进了两个对数线性形式的插值行列式方法，为算术下界和解析上界引入了新思想。
英文摘要
We prove the long-standing conjecture that, for every odd prime $p$, the only integral solutions of $a^2-2=b^p$ are $(a,b)=(\pm 1, -1)$. 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 $D \not \equiv 1 \pmod 8$. 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.
Comments44 pages, comments very welcome!
