Fermat's right triangle theorem

Fermat's right triangle theorem is a non-existence proof in number theory, the only complete proof left by Pierre de Fermat. It has several equivalent formulations:

An immediate consequence of the last of these formulations is that Fermat's last theorem is true for the exponent ${\displaystyle n=4}$.

In 1225, Fibonacci was challenged to find a construction for triples of square numbers that are equally spaced from each other, forming an arithmetic progression, and for the spacing between these numbers, which he called a congruum. One way of describing Fibonacci's solution is that the numbers to be squared are the difference of legs, hypotenuse, and sum of legs of a Pythagorean triangle, and that the congruum is four times the area of the same triangle. In his later work on the congruum problem, published in The Book of Squares, Fibonacci observed that it is impossible for a congruum to be a square number itself, but did not present a satisfactory proof of this fact.

If three squares ${\displaystyle a^{2}}$, ${\displaystyle b^{2}}$, and ${\displaystyle c^{2}}$ could form an arithmetic progression whose congruum was also a square ${\displaystyle d^{2}}$, then these numbers would satisfy the Diophantine equations

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