Congruum
In number theory, a congruum is the difference between two square numbers in an arithmetic progression of three squares. More formally, a number n is a congruum if there exist integers x, y, and z such that x2, y2, and z2 form an arithmetic progression and n = y2 - x2 = z2 - y2.
Equivalently, n is a congruum if and only if there exists a rational number x such that x2 + n and x2 - n are both squares of rational numbers.
The problem of finding congrua is closely related to the problem of finding right triangles with integer sides and a given area. Specifically, a positive integer n is a congruum if and only if it is the area of a right triangle with rational sides. Numbers that satisfy this condition are often also called congruent numbers. Determining whether a given number is a congruent number is a famously difficult problem. While efficient algorithms exist to test whether a small number is congruent, the general problem is believed to be computationally hard.