丢番图方程 $p^{x}+6^{y}=z^{2n}$：素数的一半的完全解

中文摘要 AI 辅助

设 $p$ 为素数，$n\geq 1$ 为整数。我们完全求解了正整数 $x,y,z$ 上的指数丢番图方程 $p^{x}+6^{y}=z^{2n}$，对于所有满足勒让德符号 $\leg{6}{p}$ 等于 $-1$ 的素数 $p$，即 $p\equiv 7,11,13,17\pmod{24}$：该方程无解，唯一例外是 $(p,n,x,y,z)=(13,1,1,2,7)$。由于这些剩余类在狄利克雷密度的意义下包含所有素数的一半，这就在指数 $n$ 上一致地解决了每两个素数中的一个。证明的两个要素具有独立的意义。首先，我们证明 Ramanujan--Nagell 型方程 $p^{x}=2\cdot 6^{m}+1$ 在 $x\geq 2$ 时无解，使用了 Zsigmondy 关于本原素因子的定理；这去除了作者早期关于 $n=1$ 情形的工作中出现的额外同余假设，并解决了那里提出的开放问题。其次，我们证明 $13^{x}+6^{y}=z^{2}$ 的唯一解是 $(x,y,z)=(1,2,7)$，从而完全解决了例外素数。勒让德条件是尖锐的：我们展示了对于 $\leg{6}{p}=+1$ 的素数，存在实现因式分解论证的每个分支的解。所有结果都通过广泛的计算验证得到证实，并且我们陈述了几个开放问题，包括当 $\leg{6}{p}=+1$ 时解的完全分类。

英文摘要

Let $p$ be a prime number and let $n\geq 1$ be an integer. We completely solve the exponential Diophantine equation $p^{x}+6^{y}=z^{2n}$ in positive integers $x,y,z$ for every prime $p$ such that the Legendre symbol $\leg{6}{p}$ equals $-1$, that is, for $p\equiv 7,11,13,17\pmod{24}$: the equation has no solution, with the single exception $(p,n,x,y,z)=(13,1,1,2,7)$. Since these residue classes contain half of all primes in the sense of Dirichlet density, this settles the equation for one prime out of two, uniformly in the exponent $n$. Two ingredients of the proof have independent interest. First, we show that the Ramanujan--Nagell type equation $p^{x}=2\cdot 6^{m}+1$ has no solution with $x\geq 2$, using Zsigmondy's theorem on primitive prime divisors; this removes the extra congruence hypotheses that appeared in earlier work of the author on the case $n=1$ and solves the open problems formulated there. Secondly, we prove that $13^{x}+6^{y}=z^{2}$ has $(x,y,z)=(1,2,7)$ as its unique solution, thereby completely resolving the exceptional prime. The Legendre condition is sharp: we exhibit solutions for primes with $\leg{6}{p}=+1$ realising each branch of the factorisation argument. All results are corroborated by extensive computational verification, and we state several open problems, including the complete classification of the solutions when $\leg{6}{p}=+1$.

发表机构

• University of Kara（卡拉大学）

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