Fermat's Four Squares Theorem
| dc.creator | van der Poorten, Alf | |
| dc.date | 2007-12-22 | |
| dc.date.accessioned | 2026-07-07T08:51:07Z | |
| dc.date.available | 2026-07-07T08:51:07Z | |
| dc.description | 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. | |
| dc.description | 3 pqges | |
| dc.identifier | https://arxiv.org/abs/0712.3850 | |
| dc.identifier | http://arxiv.org/abs/0712.3850 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/144829 | |
| dc.subject | Number Theory | |
| dc.subject | History and Overview | |
| dc.subject | 11-01, 11D25 | |
| dc.title | Fermat's Four Squares Theorem | |
| dc.type | text |