Generating Pythagorean Triples

Here is a neat little formula for generating Pythagorean triples, which are positive integers which could form a right triangle.

• Take any two positive integers. Call the larger m and the smaller n.
• The legs of the right triangle are given by m² - n² and 2mn, and the hypotenuse is m² + n².

For example, with m = 2 and n = 1, we get the familiar triangle with side lengths 3, 4, and 5. You can see that there is an infinite number of different Pythagorean triples, because the number of natural numbers is infinite.

Here's a quick-and-dirty Java applet to calculate some values quickly. If you're a big fan of Java applets, you may also want to check out my applet for Pascal's Art. Also visit cut-the-knot.com because they have some interesting ones.

View Source Code.

Here is a table of the first few numbers generated by this formula:

a	b	Leg X (a2 - b2)	Leg Y (2ab)	Hypotenuse (a2 + b2)	Triangle Type
					Primitive	Composite
2	1	3	4	5	P1
3	1	8	6	10		2P1
4	1	15	8	17	P2
5	1	24	10	26		2P4
6	1	35	12	37	P3
7	1	48	14	50		2P7
3	2	5	12	13	P4
4	2	12	16	20		4P1
5	2	21	20	29	P5
6	2	32	24	40		8P1
7	2	45	28	53	P6
4	3	7	24	25	P7
5	3	16	30	34		2P2
6	3	27	36	45		9P1
7	3	40	42	58		2P5
5	4	9	40	41	P8
6	4	20	48	52		4P4

See also Pythagoras.