## Friday, February 25, 2011

### Open problem about Pythagorean triangles

Definitions :

Square number
http://en.wikipedia.org/wiki/Square_number
Pythagorean Triangles and Triples
Triangular number
Pentagonal number

Triangular numbers:
http://oeis.org/A000217
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431

Pentagonal numbers: n(3n-1)/2 :
http://oeis.org/A000326
0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151

The Open problem:

It asks for Pythagorean triangles with a triangular number and a square for legs, and a pentagonal number n(3n - 1)/2 for hypothenuse.
Are there any nontrivial examples besides (3,4,5) and (105,100,145)?
Does it help to allow pentagonal numbers of negative rank, n(3n + 1)/2?