论文标题
通过复杂的持续分数找到四个正方形的总和
Finding Sums of Four Squares via Complex Continued Fractions
论文作者
论文摘要
将给定积极整数作为四个整数的总和的问题很长一段时间都广泛关注，并且对于给定的积极奇数$ n $，一个人可以通过以最大的quaternion代数进行算术来找到代表，曾经是一对（正面）整数$ x，y $ x，y $ x^2+y y $ x^2+y^2+y^2+equiv is vide。在本文中，我们介绍了一种新方法，以找到满足上述要求的$ x，y $的奇数整数$ w $的表示。该方法可以避免四元组代数中复杂的非交通性结构，该结构类似于我们用来获得质量$ p \ equiv1 \ mod4 $作为两个正方形的质量代表的代表，通过进行持续分数扩展，这是两个正方形的总和，除非我们将使用hurwitz algorithm扩展复杂的数字。
The problem of representing a given positive integer as a sum of four squares of integers has been widely concerned for a long time, and for a given positive odd $n$ one can find a representation by doing arithmetic in a maximal order of quaternion algebra once a pair of (positive) integers $x,y$ with $x^2+y^2\equiv-1\mod n$ is given. In this paper, we introduce a new method to find a representation of odd integer $w$ given $x,y$ satisfying the above requirement. This method can avoid the complicated non-commutative structure in quaternion algebra, which is similar to the one we use to obtain a representation of a prime $p\equiv1\mod4$ as sum of two squares by doing continued fraction expansions, except that here we will expand complex number using Hurwitz algorithm.
