# Pythagorean triple integers' occurrence

#### Loren

Is there a limit to how many times a particular integer may appear in different Pythagorean triples?

#### mathman

Forum Staff
The general approach is: for any pair of integers $m\gt n$, a P-triple can be defined as $a=2mn$, $b=m^2-n^2$, and $c=m^2+n^2$. Obviously there are upper limits for all three, since they are limited by the sizes of $m$ and $n$.

1 person

#### DarnItJimImAnEngineer

It wasn't that obvious to me, though upon further reflection, I think I'm starting to see that there are a finite number of m, n pairs that can produce arbitrary integer $$\displaystyle l \in \{a, b, c\}$$.

This image really helped, though. For any $l$ (at least in the range shown), it looks like you can draw a finite boundary containing triples $[a,b,c] : MIN(a,b) \leq l$ that has a relatively small number of triples within it. Even adding in the non-primitive triples isn't going to increase the number that much.

1 person