Formulas For Generating Pythagorean Triples - Dickson's Method

Dickson's Method

Leonard Eugene Dickson (1920) attributes to himself the following method for generating Pythagorean triples. To find integer solutions to, find positive integers r, s, and t such that is a square.

Then:

From this we see that is any even integer and that s and t are factors of . All Pythagorean triples may be found by this method. When s and t are coprime the triple will be primitive.

Example: Choose r = 6. Then . The three factor-pairs of 18 are: (1, 18), (2, 9), and (3, 6). All three factor pairs will produce triples using the above equations.

s = 1, t = 18 produces the triple because x = 6 + 1 = 7, y = 6 + 18 = 24, z = 6 + 1 + 18 = 25.

s = 2, t = 9 produces the triple because x = 6 + 2 = 8, y = 6 + 9 = 15, z = 6 + 2 + 9 = 17.

s = 3, t = 6 produces the triple because x = 6 + 3 = 9, y = 6 + 6 = 12, z = 6 + 3 + 6 = 15. (Since s and t are not coprime, this triple is not primitive.)