# The Congruent Number Problem: A Thousand Year Old Unsolved Problem

William Stein (http://modular.math.washington.edu)

Date: SIMUW 2006

### Abstract:

One of the oldest unsolved problems in mathematics is to determine the congruent numbers: Give a way to decide whether or not an integer is the area of a right triangle with rational side lengths. For example, 6 is the area of the right triangle with side lengths 3, 4, and 5. But 1 is not a congruent number. What about 2006? This workshop will take us from ancient algebra to a major modern conjecture in number theory that has a million dollar bounty: the Birch and Swinnerton-Dyer Conjecture.

The first few congruent numbers: 5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 101, 102, 103, 109, 110, 111, 112, 116, 117, 118, 119, 120, 124, 125, 126, 127, 133, 134, 135, 136, 137, 138, 141, 142, 143, 145, 148, 149, 150, 151, 152, 154, 156, 157, 158, 159, 161, 164, 165, 166, 167, 173, 174, 175, 180, 181, 182, 183, 184, 188, 189, 190, 191, 194, 197, 198, 199, 205, 206, 207, 208, 210, 212, 213, 214, 215, 216, 219, 220, 221, 222, 223, 224, 226, 229, 230, 231, 237, 238, 239, 240, 244, 245, 246, 247, 248, 252, 253, 254, 255, 257, 260, 261, 262, 263, 265, 269, 270, 271, 276, 277, 278, 279, 280, 284, 285, 286, 287, 291, 293, 294, 295, 299, 301, 302, 303, 306, 308, 309, 310, 311, 312, 313, 316, 317, 318, 319, 320, 323, 325, 326, 327, 330, 333, 334, 335, 336, 340, 341, 342, 343, 344, 348, 349, 350, 351, 352, 353, 357, 358, 359, 365, 366, 367, 368, 369, 371, 372, 373, 374, 375, 376, 380, 381, 382, 383, 384, 386, 389, 390, 391, 395, 397, 398, 399, 404, 405, 406, 407, 408, 410, 412, 413, 414, 415, 421, 422, 423, 426, 429, 430, 431, 434, 436, 437, 438, 439, 440, 442, 444, 445, 446, 447, 448, 453, 454, 455, 457, 461, 462, 463, 464, 465, 468, 469, 470, 471, 472, 476, 477, 478, 479, 480, 485, 486, 487, 493, 494, 495, 496, ..., 1974, 1975, 1976, 1980, 1981, 1982, 1983, 1984, 1989, 1990, 1991, 1995, 1997, 1998, 1999, 2000, 2004, 2005, 2006, 2007, 2008, 2009,...

William Stein 2006-07-07