Pythagorean triple integers' occurrence

May 2015
516
32
Arlington, VA
Is there a limit to how many times a particular integer may appear in different Pythagorean triples?
 

mathman

Forum Staff
May 2007
6,933
775
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 $n$.
 
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Jun 2019
493
262
USA
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.
 
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