Congruum

In number theory, a congruum (plural congrua) is the difference between successive square numbers in an arithmetic progression of three squares. The congruum problem is the problem of finding squares in arithmetic progression and their associated congrua. It can be formalized as a Diophantine equation.

Fibonacci solved the congruum problem by finding a parameterized formula for generating all congrua, together with their associated arithmetic progressions. According to this formula, each congruum is four times the area of a Pythagorean triangle, a right triangle whose sides are integers. Congrua are also closely connected with congruent numbers, the areas of right triangles whose sides are rational numbers. Every congruum is a congruent number, and every congruent number is a congruum multiplied by the square of a rational number.

Fibonacci claimed without proof that it is impossible for a congruum to be a square number. This was later proven by Pierre de Fermat as Fermat's right triangle theorem.