特殊数与不定方程

考什么 · What it tests

需要先会 · Prerequisites

• 质因数分解 / Prime factorization
• 完全平方数、阶乘的定义 / The definitions of perfect squares and factorials
• 整除与因数 / Divisibility and factors

常见套路 · Common moves

1. 不定方程靠范围 + 整除收网：一个方程多个未知数，先用「正整数」「不超过某值」把范围压小，再逐个验，答案是有限的——这类题的通用套路。

   Corner a Diophantine equation with ranges and divisibility: one equation, several unknowns — shrink the range with "positive integer" and upper bounds, then test each; the answers are finite.
2. 连续整数和拆成「中间数 × 个数」：奇数个连续整数的和 = 正中间那个 × 个数，偶数个则用首尾配对。

   Split a sum of consecutive integers as "middle × count": an odd count sums to the middle term times the count; an even count pairs the ends.
3. 完全平方数看质因数指数：$n$ 是完全平方数当且仅当质因数分解里每个指数都是偶数。

   Judge a perfect square by exponents: $n$ is a perfect square exactly when every exponent in its prime factorization is even.

易错点 · Common pitfalls

• 枚举整数解时漏掉边界或负数/零：题目没说正整数时，0 和负数也可能是解。

  Missing the boundary or negatives/zero when enumerating: unless the problem says positive integers, 0 and negatives may be solutions too.
• 把「连续整数」默认成从 1 开始：题目常是任意一段连续整数，别自动从 1 起。

  Assuming "consecutive integers" start at 1: they're usually any run of consecutive integers, so don't default to starting at 1.