AI 中文总结
该研究探讨广义费马方程 $Ax^\ell + By^m + Cz^n = 0$ 的 $p$-进可解性，推导其局部可解比例规律，给出两两互素与非两两互素情形的结论并辅以实例说明。
AI 中文摘要
我们研究正整数 $\ell,m,n$ 对应的广义费马方程 $Ax^\ell + By^m + Cz^n = 0$ 的 $p$-进数域可解性。对除有限个素数 $p$ 外的所有素数，存在 $p$-进解的概率由仅依赖 $\gcd(p-1,\ell,m)$、$\gcd(p-1,\ell,n)$ 和 $\gcd(p-1,m,n)$ 的 $p$ 的有理函数描述。当 $\ell,m,n$ 两两互素时，我们推导得出，这些方程在所有局部均可解的比例为正，等于这些局部概率的乘积；当 $\ell,m,n$ 非两两互素时，该比例为 0%。随后我们给出若干详细例子以说明结果的具体性。
英文摘要
We study $p$-adic solubility of generalized Fermat equations $Ax^\ell + By^m + Cz^n = 0$ for positive integers $\ell,m,n$. For all but finitely many primes $p$, the probability of having a $p$-adic solution is described by a rational function in $p$ depending only on $\gcd(p-1,\ell,m)$, $\gcd(p-1,\ell,n)$, and $\gcd(p-1,m,n)$. When $\ell,m,n$ are pairwise coprime, we deduce that the proportion of these equations which are everywhere locally soluble is positive, given by a product of these local probabilities; when $\ell,m,n$ are not pairwise coprime, the proportion is 0\%. We then give several detailed examples demonstrating the explicit nature of the results.
CommentsMain article by Keyes and Kobin; joint appendix with Arango-Piñeros
