初等数论在正交拉丁方构造中的应用

Application of Elementary Number Theory to Constructions of Orthogonal Latin Squares

摘要: 初等数论是数学专业的一门基础课程，学生在学习同余、可逆元和同余方程后，往往难以认识这些知识的应用。本文以拉丁方和正交拉丁方为例，介绍利用模运算构造循环拉丁方和一般线性拉丁方，并利用一次同余方程的唯一可解性判定两个线性拉丁方的正交性。通过这些经典构造，说明初等数论知识在组合设计中的具体应用，引导学生理解同余理论的结构意义和应用价值。

Abstract: Elementary number theory is a fundamental course for mathematics majors, but students often find it difficult to recognize the applications of congruences, invertible elements, and congruence equations after learning these concepts. Taking Latin squares and orthogonal Latin squares as examples, this paper introduces the construction of cyclic Latin squares and general linear Latin squares using modular arithmetic, and presents a criterion for the orthogonality of two linear Latin squares based on the unique solvability of linear congruence equations. These classical constructions illustrate concrete applications of elementary number theory in combinatorial design and help students understand the structural significance and application value of congruence theory.