Fermat's Four Squares Theorem

Details Fermat's Four Squares Theorem Fermat's Four Squares Theorem

2007-12-22

arxiv.org/abs/0712.3850v1

Mathematics

Number Theory

3 pqges

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Fermat's Four Squares Theorem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Fermat's Four Squares Theorem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Fermat's Four Squares Theorem will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-425891