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 x2, y2, and z2 (for integers x, y, and z) are three square numbers that are equally spaced apart from each other, then the spacing between them, z2 − y2 = y2 − x2, 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 x, y, and 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.
For instance, the number 96 is a congruum, since it is the difference between each pair of the three squares 4, 100, and 196 (the squares of 2, 10, and 14 respectively).
The first few congrua are:
The congruum problem was originally posed in 1225, as part of a mathematical tournament held by Frederick II, Holy Roman Emperor, and answered correctly at that time by Fibonacci, who recorded his work on this problem in his Book of Squares.
Fibonacci was already aware that it is impossible for a congruum to itself be a square, but did not give a satisfactory proof of this fact. Geometrically, this means that it is not possible for the pair of legs of a Pythagorean triangle to be the leg and hypotenuse of another Pythagorean triangle. A proof was eventually given by Pierre de Fermat, and the result is now known as Fermat's right triangle theorem. Fermat also conjectured, and Leonhard Euler proved, that there is no sequence of four squares in arithmetic progression.
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