Pythagorean Triples

Below is a listing of Pythagorean triples, integers (a,b,c) such that a^2 + b^2 = c^2. I was intrigued by them in high school, because certain math competitions used them and you could save yourself the time and trouble of computing a square root if you memorized a few common ones. (3,4,5) was very common; they'd often put in (8,15,17) because it was less well-known, but still computable with pencil and paper.

I have only listed "primitive" (or relatively prime) triples having no common divisors. Thus, (6,8,10) is not listed, because it is a multiple of (3,4,5). I have also only searched for triples where the first two numbers are less than 400. You can compile and run my program yourself and give a larger number on the command line (or see this Java applet based on my code, which goes up to 5000).

I want to call attention to some patterns.

• Since each odd number is the the difference of squares of adjacent numbers (1=1-0, 3=4-1, 5=9-4, 7=16-9, 9=25-16, ...), each odd perfect square will be the difference of two adjacent squares (9=25-16, 25=169-144, ...), so that every odd number will be part of a triple (o, n, n+1).
• Another pattern involves multiples of 4 and two adjacent odd numbers. (Note that 3,4,5 fits this pattern as well.)

The University of Cambridge had an interesting theorem by A.F. Beardon (alas, link is now dead) that shows that all triples are of the form

I recently came across a new link about Pythagorean triples.

- Geoffrey Coram

  gjcoram(at)yahoo(dot)com

A few examples of each pattern:

Pattern 1	Pattern 2	other pattern ?
3, 4, 5
5, 12, 13
7, 24, 25	8, 15, 17
9, 40, 41
11, 60, 61	12, 35, 37
13, 84, 85
15, 112, 113	16, 63, 65
17, 144, 145
19, 180, 181	20, 99, 101	20, 21, 29
21, 220, 221
23, 264, 265	24, 143, 145
25, 312, 313
27, 364, 365	28, 195, 197	28, 45, 53
... (b>400)	32, 255, 257	33, 56, 65
	36, 323, 325	36, 77, 85

The list in numerical order:

3, 4, 5
5, 12, 13
7, 24, 25
8, 15, 17
9, 40, 41
11, 60, 61
12, 35, 37
13, 84, 85
15, 112, 113
16, 63, 65
17, 144, 145
19, 180, 181
20, 21, 29
20, 99, 101
21, 220, 221
23, 264, 265
24, 143, 145
25, 312, 313
27, 364, 365
28, 45, 53
28, 195, 197
32, 255, 257
33, 56, 65
36, 77, 85
36, 323, 325
39, 80, 89
40, 399, 401
44, 117, 125
48, 55, 73
51, 140, 149
52, 165, 173
57, 176, 185
60, 91, 109
60, 221, 229
65, 72, 97
68, 285, 293
69, 260, 269
75, 308, 317
76, 357, 365
84, 187, 205
85, 132, 157
88, 105, 137
95, 168, 193
96, 247, 265
104, 153, 185
105, 208, 233
115, 252, 277
119, 120, 169
120, 209, 241
120, 391, 409
133, 156, 205
135, 352, 377
136, 273, 305
140, 171, 221
152, 345, 377
160, 231, 281
161, 240, 289
175, 288, 337
180, 299, 349
189, 340, 389
203, 396, 445
204, 253, 325
207, 224, 305
225, 272, 353
228, 325, 397
252, 275, 373
261, 380, 461
280, 351, 449
297, 304, 425
319, 360, 481
336, 377, 505