Fermat's Four Squares Theorem

Abstract: It is easy to find a right-angled triangle with integer sides whose area is 6. There is no such triangle with area 5, but there is one with rational sides (a \emph{Pythagorean triangle}'). For historical reasons, integers such as 6 or 5 that are (the squarefree part of) the area of some Pythagorean triangle are called\emph{congruent numbers}'. These numbers actually are interesting for the following reason: Notice the sequence $\frac14$, $6\frac14$, $12\frac14$. It is an arithmetic progression with common difference 6, consisting of squares $(\frac12)^2$, $(\frac52)^2$, $(\frac72)^2$ of rational numbers. Indeed the common difference of three rational squares in AP is a congruent number and every congruent number is the common difference of three rational squares in arithmetic progression. The triangle given by $9^{2}+40^{2}=41^{2}$ has area $180=5\cdot6^{2}$ and the numbers $x-5$, $x$ and $x+5$ all are rational squares if $x=11{97/144}$. Recall one obtains all Pythagorean triangles with relatively prime integer sides by taking $x=4uv$, $y=\pm(4u^{2}-v^{2})$, $z=4u^{2}+v^{2}$ where $u$ and $v$ are integers with $2u$ and $v$ relatively prime. Fermat proved that there is no AP of more than three squares of rationals.