We develop machinery to explicitly determine, in many instances, when the difference $x^2-y^n$ is divisible only by powers of a given fixed prime. This combines a wide variety of techniques from Diophantine approximation (bounds for linear forms in logarithms, both archimedean and nonarchimedean, lattice basis reduction, methods for solving Thue–Mahler and $S$-unit equations, and the primitive divisor theorem of Bilu, Hanrot and Voutier) and classical algebraic number theory, with results derived from the modularity of Galois representations attached to Frey–Hellegoaurch elliptic curves. By way of example, we completely solve the equation

where $2\le q<100$ is prime, and $x,y,\alpha$ and $n$ are integers with $n\ge 3$ and $\gcd <!--FUNCTION\; APPLICATION-->(x,y)=1$.

在许多情况下，我们开发了一种机制来明确确定差异何时出现$X^2-y^n$只能被给定固定素数的幂整除。它结合了丢番图近似（对数线性形式的界限，阿基米德和非阿基米德、格基约简、求解 Thue-Mahler 的方法和$S$-单位方程，以及 Bilu、Hanrot 和 Voutier 的原除数定理）和经典代数数论，其结果源自附加到 Frey-Hellegoaurch 椭圆曲线的 Galois 表示的模块化。举例来说，我们完全求解方程

在哪里$2\le q<100$是素数，并且$X,y,\alpha$和$n$是整数$n\ge 3$和$\mathrm{最大CD}<!--FUNCTION\; APPLICATION-->（X,y）=1$。