Pythagorean Triples

Pythagorean Triples are sets of three integers for which the statement, A² + B² = C², is true. This is a special case of the Pythagorean Theorem which states for all right triangles, A² + B² = C². While the most common introduction of the Pythagorean Theorem to students uses the triangle with sides, A=3, B=4, C=5, right triangles are not limited to sides of integer length. The Pythagorean Triple for this triangle is (3,4,5). An example of a right triangle with non-integers sides is represented by the triangle form by the two sides and the diagonal of an 8.5 x 11 inch sheet of paper. This triangle has sides of 8.5, 11 and 13.901 inches (13.901 is the square root of the sum of 8.5 square and 11 square).

Other Pythagorean triples include (5,12,13), (8,15,17) and (7,24,25). There are an infinite number of Pythagorean Triples.

I recently came across a formula to generate additional Triples.

Let n and m be any positive integers where n>m. The Pythagorean Triple generated is formed by the numbers:
2nm
n² - m²
n² +m²

For example, starting with n=2 and m=1, one generates the (3.4,5) Triple.

2nm = 4
n² - m² = 3
n² +m² = 5

Another example, with n=3, n=2, generates the (5,12,13) Triple

2nm = 12
n² - m² = 5
n² +m² = 13

I wrote an Excel spreadsheet to generate Triples up to n=12, m=11 and produced the following:

N	M		A	B	C
2	1		3	4	5
3	1		8	6	10
3	2		5	12	13
4	1		15	8	17
4	2		12	16	20
4	3		7	24	25
5	1		24	10	26
5	2		21	20	29
5	3		16	30	34
5	4		9	40	41
6	1		35	12	37
6	2		32	24	40
6	3		27	36	45
6	4		20	48	52
6	5		11	60	61
7	1		48	14	50
7	2		45	28	53
7	3		40	42	58
7	4		33	56	65
7	5		24	70	74
7	6		13	84	85
8	1		63	16	65
8	2		60	32	68
8	3		55	48	73
8	4		48	64	80
8	5		39	80	89
8	6		28	96	100
8	7		15	112	113
9	1		80	18	82
9	2		77	36	85
9	3		72	54	90
9	4		65	72	97
9	5		56	90	106
9	6		45	108	117
9	7		32	126	130
9	8		17	144	145
10	1		99	20	101
10	2		96	40	104
10	3		91	60	109
10	4		84	80	116
10	5		75	100	125
10	6		64	120	136
10	7		51	140	149
10	8		36	160	164
10	9		19	180	181
11	1		120	22	122
11	2		117	44	125
11	3		112	66	130
11	4		105	88	137
11	5		96	110	146
11	6		85	132	157
11	7		72	154	170
11	8		57	176	185
11	9		40	198	202
11	10		21	220	221
12	1		143	24	145
12	2		140	48	148
12	3		135	72	153
12	4		128	96	160
12	5		119	120	169
12	6		108	144	180
12	7		95	168	193
12	8		80	192	208
12	9		63	216	225
12	10		44	240	244
12	11		23	264	265

In the table above, while C is always the longest leg, either A or B can be the shortest side of the triangle.

Also, certain Triples are highlighted yellow. These Triples are unique in that they produce triangles that are not similar to triangles produced by any prior set of Triples. For example, the set (12,16,20) has sides all divisible by 4 so it produces a triangle that is similar to (3,4,5). While the set (8,15,17) is highlighted in that the three numbers has greatest common denominator (GCD) of 1. Therefore, its triangle is not similar to the triangle of any prior set of Triples.

Generator:
Pythagorean Triple Generator
Update 9/24/2022 - see post Math Vacation: Pythagorean Triple Generators - Part III (jamesmacmath.blogspot.com) for additional methods of producing Pythagorean Triples. The generator spreadsheet (above) has been updated with these additional methods.