Pythagorean triple

Pythagorean triple is a triple of integers that follow the following formula:

$a^{2}+b^{2}=c^{2}$

It is similar to the Pythagorean Theorem which states the same for a right triangle for non-integer sides. Some of the simplest triples are:

$3^{2}+4^{2}=5^{2}$ and $5^{2}+12^{2}=13^{2}$

Studying these two triples, the reader may have the insight that

$3^{2}=4+5$ and $5^{2}=12+13$

This leads to a trivial sequence that generates some but not all possible triples is based on the positive integers starting with $a=3$::

If $a$ is odd, then $b={\frac {a^{2}}{2}}-{\frac {1}{2}}$ and $c=b+1$

If $a$ is even, then $b={\frac {a^{2}}{4}}-1$ and $c=b+2$

Euclid's formula is a fundamental formula for generating all possible Pythagorean triples given an arbitrary pair of integers $m$ and $n$ with $m>n>0$. The formula states that the following integers form a Pythagorean triple:

$a=m^{2}-n^{2},\ \,b=2mn,\ \,c=m^{2}+n^{2}$

There are many other formulas for generating such triples.