Congruum

In number theory, a congruum (plural congrua) is the difference between successive square numbers in an arithmetic progression of three squares. That is, if ${\displaystyle x^{2}}$, ${\displaystyle y^{2}}$, and ${\displaystyle z^{2}}$ (for integers ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$) are three square numbers that are equally spaced apart from each other, then the spacing between them, ${\displaystyle z^{2}-y^{2}=y^{2}-x^{2}}$, is called a congruum.

The congruum problem is the problem of finding squares in arithmetic progression and their associated congrua. It can be formalized as a Diophantine equation: find integers ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ such that

When this equation is satisfied, both sides of the equation equal the congruum.

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. Congrua are also closely connected with congruent numbers: every congruum is a congruent number, and every congruent number is a congruum multiplied by the square of a rational number.