素数幂丢番图数组
Prime-power Diophantine tuples
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中文总结 AI 辅助
该研究将丢番图数组的间隙论证推广到素数幂情形，得到正约化 $D(\pm p^r)$ 数组的基数一致上界，明确了不同素数幂下的基数界限及最大基数的阶性质。
中文摘要 AI 辅助
正 $D(n)$-$m$-数组是指由 $m$ 个不同正整数组成的集合 $A=\{a_1,\ldots,a_m\}$，满足对任意 $i\ne j$，$a_i a_j+n$ 为平方数。2005年，Dujella和Luca得到了正整数的 $D(p)$ 或 $D(-p)$ 数组的基数的绝对界，该界对素数 $p$ 是一致的。我们将基础间隙论证推广到素数幂情形。通过初等线性代数证明的显式环面消去证书取代了未饱和齐次消去步骤，且对每个素数幂模均有效。该消去式的显式次数和高度界给出了一致间隙原理。结合有界 $|n|$ 的一般界，这表明每个正的约化 $D(\pm p^r)$ 数组（即元素不被 $p$ 整除的数组）的元素个数小于 $2^{121}$，与 $p$ 和 $r$ 无关。因此，每个正 $D(\pm p)$ 数组最多有 $2^{121}$ 个元素，正 $D(\pm p^2)$ 数组的元素个数小于 $2^{122}$，且正 $D(\pm p^r)$ 数组的最大基数关于 $p$ 是一致的 $O(r)$。
英文摘要
A positive $D(n)$-$m$-tuple is a set $A=\{a_1,\ldots,a_m\}$ of distinct positive integers such that $a_i a_j+n$ is a square for every $i\ne j$. In 2005, Dujella and Luca obtained an absolute bound for the cardinality of a $D(p)$- or $D(-p)$-tuple of positive integers, uniformly in the prime $p$. We extend the underlying gap argument to prime powers. An explicit toric elimination certificate, proved by elementary linear algebra, replaces the unsaturated homogeneous elimination step and is valid modulo every prime power. Explicit degree and height bounds for this eliminant yield a uniform gap principle. Combined with the general bound for bounded $|n|$, this shows that every positive reduced $D(\pm p^r)$-tuple (i.e. tuple with elements not divisible by $p$) has less than $2^{121}$ elements, independently of $p$ and $r$. Thus, every positive $D(\pm p)$-tuple has at most $2^{121}$ elements, positive $D(\pm p^2)$-tuples have less than $2^{122}$ elements, and the maximal cardinality of a positive $D(\pm p^r)$-tuple is $O(r)$ uniformly in $p$.
