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 divisors ${\rm {gcd}}(a,b)={\rm {gcd}}(a,c)={\rm {gcd}}(b,c)=1$. Because of their relationship through the Pythagorean theorem, a, b, and c are coprime if a and b are coprime (${\rm {gcd}}(a,b)=1$). Each triple forms 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 credit: 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>

Java

Theoretically, this can go "forever", but it takes a while, so only the minimum is shown. Luckily, BigInteger has a GCD method built in.

<lang java>import java.math.BigInteger;

public class PythTrip{ public static void main(String[] args){ long tripCount = 0, primCount = 0;

//change this to whatever perimeter limit you want;the RAM's the limit BigInteger periLimit = new BigInteger("100");

for(BigInteger a = BigInteger.ONE; a.compareTo(periLimit) < 0; a = a.add(BigInteger.ONE)){ for(BigInteger b = new BigInteger(a.toString()); b.compareTo(periLimit) < 0; b = b.add(BigInteger.ONE)){ for(BigInteger c = new BigInteger(b.toString()); c.compareTo(periLimit) < 0; c = c.add(BigInteger.ONE)){ if(a.add(b).add(c).compareTo(periLimit) > 0)break; if(a.pow(2).add(b.pow(2)).compareTo(c.pow(2)) != 0) continue;

tripCount++; System.out.print(a +", "+b+", "+c);

//does binary GCD under the hood if(a.gcd(b).compareTo(BigInteger.ONE) == 0){ System.out.print(" primitive"); primCount++; } System.out.println(); } } } System.out.println("Up to a perimeter of "+periLimit+", there are "+ tripCount+" triples, of which "+primCount+" are primitive."); } }</lang> Output:

3, 4, 5 primitive
5, 12, 13 primitive
6, 8, 10
7, 24, 25 primitive
8, 15, 17 primitive
9, 12, 15
9, 40, 41 primitive
10, 24, 26
12, 16, 20
12, 35, 37 primitive
15, 20, 25
15, 36, 39
16, 30, 34
18, 24, 30
20, 21, 29 primitive
21, 28, 35
24, 32, 40
Up to a perimeter of 100, there are 17 triples, of which 7 are primitive.