V2EX relatively prime

Relatively Prime

Definition / 定义

(数论)互素的:指两个整数除了 1 以外没有任何共同的正因数;等价地说,它们的最大公因数(gcd)为 1。也常用同义词 coprime

Pronunciation / 发音

/rltvli pram/

Examples / 例句

12 and 25 are relatively prime.
12 和 25 互素。

If \(a\) and \(b\) are relatively prime, then any common divisor of \(a\) and \(bc\) must also divide \(c\).
如果 \(a\) 与 \(b\) 互素,那么 \(a\) 与 \(bc\) 的任何公因数也一定是 \(c\) 的因数。

Etymology / 词源

relatively 源自 relative(“相对的”),表示“相对于某个关系而言”;prime 在数学语境里与“质数/基本的”相关。合起来的意思是“在因数关系上相对独立”,即两数之间不共享除 1 以外的因子。该术语在英语数学写作中常与 coprime 并用。

Related Words / 相关词汇

Literary Works / 文学作品

  • Euclid, Elements(《几何原本》;虽然古典文本不一定用英语短语 “relatively prime”,但系统讨论了互素与公因数思想)
  • G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers(《数论导引》;常见 “coprime/relatively prime” 相关表述)
  • Tom M. Apostol, Introduction to Analytic Number Theory(《解析数论导论》;大量使用互素条件与相关定理)
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