Pythagorean triples

Pythagorean triples is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

A Pythagorean triple is defined as three positive integers $(a,b,c)$ where $a\leq b\leq c$, and $a^{2}+b^{2}=c^{2}.$ They are called primitive triples if $a,b,c$ are coprime, that is, if their pairwise greatest common denominators ${\rm {gcd}}(a,b)={\rm {gcd}}(a,c)={\rm {gcd}}(b,c)=1$. Each triple form the length of the sides of a right triangle, whose perimeter is $P=a+b+c$.

Task

How many Pythagorean triples are there with a perimeter no larger than 100? Of these, how many are primitive?

Extra: Can your program handle a max perimeter of 1,000,000? What about 10,000,000? 100,000,000?

C

Sample implemention; naive method, patently won't scale to larger numbers. <lang C>#include <stdio.h>

int gcd(int m, int n) { int t; while (n) { t = n; n = m % n; m = t; } return m; }

int main() { int a, b, c; int pytha = 0, prim = 0; for (a = 1; a < 100; a++) { for (b = a; b < 100; b++) { for (c = b; c < 100; c++) { if (a + b + c > 100) break; if (a * a + b * b != c * c) continue;

pytha++; if (gcd(a, b) == 1) prim++; } } }

printf("Up to 100, there are %d triples, of which %d are primitive\n", pytha, prim); return 0; }</lang>output:<lang>Up to 100, there are 17 triples, of which 7 are primitive</lang>