素数三角定理：二元二次型与本原勾股三元组
The Triangular Theorem of the Primes: Binary Quadratic Forms and Primitive Pythagorean Triples
摘要：
本文报道了原始毕达哥拉斯三角形中二元二次型的出现及其几何解释。除了众所周知的事实，即边长为整数的直角三角形的边长z（互素）长度，可以表示为两个平方的和，z=a^2+b^2，其中a和b是具有相反奇偶性的正整数，使得a>b>0且gcd（a，b）=1，表明两边x和y的和，也可以表示为二元二次型，x+y=（a+b）^2-2b^2。类似地，当考虑到内切圆的半径r=b（a-b）时，可以得到第三个二元二次型，即（x+y）-4r=z-2 r =（a-b）^2+2b^2。这三种二次表示法适用于正整数，它们的因子分解只能包括由相同类型的二进制二次型表示的素数p，即分别为p= 1，5（mod 8）、p= 1，7（mod 8）和p= 1，3（mod 8）。对于所有三种类型的二元二次型，当表示的正整数是素数时，这样的表示是唯一的。这意味着所有奇素数都可以几何地合并到原始毕达哥拉斯三角形中。
英文摘要：
This article reports the occurrence of binary quadratic forms in primitive Pythagorean triangles and their geometric interpretation. In addition to the well-known fact that the hypotenuse, z, of a right triangle, with sides of integral (relatively prime) length, can be expressed as the sum of two squares, z=a^2+b^2, where a and b are positive integers of opposite parity such that a>b>0 and gcd(a,b)=1, it is shown that the sum of the two sides, x and y, can also be expressed as a binary quadratic form, x+y=(a+b)^2-2b^2. Similarly, when the radius of the inscribed circle is taken into account, r=b(a-b), a third binary quadratic form is found, namely (x+y)-4r=z-2r=(a-b)^2+2b^2. The three quadratic representations accommodate positive integers whose factorizations can only include primes p represented by the same type of binary quadratic forms, i.e. p=1,5(mod8), p=1,7(mod8), and p=1,3(mod8), respectively. For all three types of binary quadratic forms, when the positive integers represented are prime, such representations are unique. This implies that all odd primes can be geometrically incorporated into primitive Pythagorean triangles.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
