Hello, MATNTRNG!
I have a very primitive solution . . .
All Pythagorean triples are generated by: .Find the only Pythagorean triplet, , for which
We have: .
. . which simplifies to: .
There are only seven solutions.
. .
And only the last triplet has a sum of 1000.
The proof of a=m^2-n^2,b=2mn,c=m^2+n^2 assumed that (a,b)=1. Although it's easily seen that the formula still applies if (a,b) is a perfect square, but there's a potential that you'll miss some solution (although here it doesn't happen) if you merely consider a=m^2-n^2,b=2mn