[seqfan] Re: [math-fun] Triangular+Triangular = Factorial

[Douglas McNeil]

> probably relatively moderate amount of elliptic curve factoring will
> pick up the low hanging fruit.

Seemed straightforward enough.  I can now exclude z values of

excluded: [6, 11, 12, 14, 18, 19, 20, 22, 23, 25, 26, 30, 31, 35, 36,
37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56,
57, 58, 60, 61, 62, 63, 64, 65, 67, 70, 71, 73, 75, 76, 77, 78, 82,
83, 84, 87, 88, 89, 90, 91, 92, 94, 97, 98, 101, 103, 104, 105, 106,
107, 108, 110, 111, 112, 113, 115, 116, 117, 119, 120, 121, 122, 123,
124, 127, 128, 130, 132, 136, 137, 138, 141, 142, 144, 145, 146, 147,
148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161,
163, 166, 167, 168, 169, 170, 172, 174, 175, 176, 177, 179, 180, 181,
182, 183, 185, 186, 187, 188, 191, 193, 194, 195, 196, 197, 198, 200,
201, 202, 203, 204, 205, 206, 207, 210, 211, 213, 214, 215, 216, 218,
223, 225, 226, 227, 228, 229, 230, 231, 232, 236, 237, 238, 239, 240,
241, 243, 245, 246, 247, 249, 250, 251, 253, 255, 256, 257, 258, 260,
261, 262, 263, 265, 266, 267, 269, 270, 271, 272, 273, 274, 276, 278,
279, 281, 282, 283, 284, 285, 287, 288, 289, 290, 292, 293, 295, 296,
297, 298, 300, 301, 302, 303, 304, 306, 307, 309, 311, 313, 314, 315,
316, 317, 319, 320, 321, 322, 324, 327, 331, 333, 334, 335, 336, 338,
339, 340, 343, 344, 346, 348, 350, 352, 353, 354, 355, 356, 358, 359,
360, 362, 365, 366, 367, 368, 369, 370, 372, 373, 379, 380, 381, 382,
383, 384, 386, 387, 388, 389, 390, 394, 395, 398, 399, 400, 401, 403,
404, 405, 406, 407, 408, 412, 413, 414, 415, 416, 417, 418, 419, 420,
421, 423, 424, 425, 426, 427, 429, 432, 433, 436, 437, 438, 439, 440,
441, 443, 444, 445, 446, 447, 449, 450, 451, 452, 454, 456, 458, 459,
460, 462, 464, 469, 470, 472, 474, 479, 486, 487, 488, 489]

and can fully factor 8*(z!)+2 for

fully factored: [1, 2, 3, 4, 5, 7, 8, 9, 10, 13, 15, 16, 17, 21, 24,
27, 28, 29, 32, 33, 34, 42, 49, 54, 59, 66, 68, 72, 79, 85, 86, 95,
96, 102, 118, 129, 135, 164, 184, 190, 219, 221, 264, 351, 357, 457,
466]

This leaves only 7 numbers, [69, 74, 80, 81, 93, 99, 100] <= 100 as
undecided (123 in total < 500) but they could be within reach.  I set
a pretty short time limit on the ecm runs, to get a first pass done
overnight.  Anyway, it should be complete up to z <= 68.

Note that there are two differences with results in this thread.
Trivial: 2 I think was put in by accident in the previous exclusion
list (8*(2!)+2 = 18 = 9+9); I agree with everything else.
Non-trivial: the results quoted by R.K. Guy listed no solution for
z=59, but 8*(59!)+2 has prime factors

[2, 8249057, 2174500369, 67820061593, 114446404287889,
11247609217977437, 354241345536913447681853]

in which all factors 3 mod 4 have even exponent (namely, 0).


Doug

-- 
Department of Earth Sciences
University of Hong Kong