AI 中文总结
本文研究由三角数构成的D(n)丢番图元组，证明不存在D(1)三角四元组，提出生成D(1)三角三元组的算法，利用广义佩尔方程设计高效判定算法，还得出相关数论结论并推进了判据研究。
AI 中文摘要
具有D(n)性质的m元组是指由m个正整数(a₁,a₂,…,aₘ)组成的元组，满足对所有1≤i<j≤m，aᵢaⱼ + n均为平方数。第k个三角数为Tₖ = k(k+1)/2，其中k为非负整数。本文仅考虑由三角数构成的D(1)元组，证明不存在任何D(1)三角四元组，并描述一种算法以生成无限族的D(1)三角三元组，推测该无限族包含所有D(1)三角三元组。此外，本文还研究一般的D(n)元组。为解决计算难题，本文提出一种利用广义佩尔方程（Generalized Pell Equations, GPE）的高效算法，用于判定Tₐ是否属于某D(n)三角对，该算法的时间复杂度为O(a^(1/2))。随后本文证明，当n≡2,5 (mod 9)时，不存在D(n)三角对，并讨论其他似乎不存在D(n)三角对的n值。本文还证明，对于所有p≠3，该D(n)方程在所有p进数域ℚₚ中均有解，最后呈现了确定不存在D(n)三角对的n的一般判据的研究进展。
英文摘要
A $m$-tuple with the property $D(n)$ is a tuple of $m$ positive integers $(a_1, a_2, \dots, a_m)$ such that $a_i a_j + n$ is an square, for $1 \le i < j \le m$. The $k$th triangular number is $T_k = \frac{k(k+1)}{2}$ for nonnegative integers $k$. We consider $D(1)$ tuples consisting only of triangular numbers. We prove the nonexistence of any $D(1)$ triangular quadruple and describe an algorithm to generate an infinite family of $D(1)$ triangular triples, which we conjecture contains all $D(1)$ triangular triples. We also consider general $D(n)$ tuples. To aid with computational difficulties, we present an efficient algorithm, using Generalized Pell Equations (GPEs), to determine whether $T_a$ is in a $D(n)$ triangular pair, which runs in $O(a^{1/2})$ time. We then prove that no $D(n)$ triangular pair exists for $n \equiv 2,5 \text{ (mod } 9\text{)}$, and discuss other values of $n$ for which there appear to be no $D(n)$ triangular pairs. We also show that our $D(n)$ equation has solutions in all $\mathbb{Q}_p$, for $p \neq 3$. We then present progress on determining a general criteria on $n$ for which no $D(n)$ triangular pairs exist.
Comments15 pages, 0 figures, abstract presented at 2025 JMM in PME Contributed Session on Research by Undergraduates, VIII
