Pythagorean triples

Pythagorean triples consists of three positive integers a, b, and c, such that a2 + b2 = c2. These triples are commonly written as (a, b, c), and a typical example is (3, 4, 5); 32 + 42 = 52 or 9 + 16 = 25. A primitive Pythagorean triple is one in which a, b and c are coprime (gcd(a, b, c) = 1) and for any primitive Pythagorean triple, (ka, kb, kc) for any positive integer k is a non-primitive Pythagorean triple. Euclid's formula (300 BC) will generate Pythagorean triples given an arbitrary pair of positive integers m and n with m > n > 0. A primitive Pythagorean triple additionally require:
a=m^2-n^2, b=2mn \mbox{ and } c=m^2+n^2
It's easy to check algebraically that the sum of the squares of the first two is the same as the square of the last one. Below is a table of [a, b, c] for n between 1 and 9 and m between 2 and 10. The highlighted cells show primitive Pythagorean triples (GCD(a, b, c) = 1).
n =123456789
m ↓
2[3,4,5]
3[8,6,10][5,12,13]
4[15,8,17][12,16,20][7,24,25]
5[24,10,26][21,20,29][16,30,34][9,40,41]
6[35,12,37][32,24,40][27,36,45][20,48,52][11,60,61]
7[48,14,50][45,28,53][40,42,58][33,56,65][24,70,74][13,84,85]
8[63,16,65][60,32,68][55,48,73][48,64,80][39,80,89][28,96,100][15,112,113]
9[80,18,82][77,36,85][72,54,90][65,72,97][56,90,106][45,108,117][32,126,130][17,144,145]
10[99,20,101][96,40,104][91,60,109][84,80,116][75,100,125][64,120,136][51,140,149][36,160,164][19,180,181]
Euclid of AlexandriaEuclid of Alexandria was one of the most prominent mathematicians of antiquity best known for his treatise on mathematics, The Elements.