The Triangular Theorem of the Primes: Binary Quadratic Forms and Primitive Pythagorean Triples

TL;DR

This paper explores the relationship between binary quadratic forms and primitive Pythagorean triangles, revealing new geometric interpretations and prime representations through quadratic forms.

Contribution

It introduces novel binary quadratic form representations for the sides and properties of primitive Pythagorean triangles, linking prime factorizations to geometric structures.

Findings

Hypotenuse expressed as sum of two squares, z=a^2+b^2.

Sum of triangle sides as a quadratic form, x+y=(a+b)^2-2b^2.

Prime representations are unique and linked to specific quadratic forms.

Abstract

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.