witness #44533

N: 44,455
|A|: 170
exponent log |A| / log N: 0.479881

status: current record for this modulus
submitted by: Rayan Hatout
submitted at: 2026-09-01 09:13:57 UTC

Elements (170)

546, 577, 631, 662, 1103, 1188, 3189, 3274, 4205, 4239, 4290, 4324, 4726, 4727, 4811, 4812, 5804, 5889, 6287, 6372, 7340, 7374, 7425, 7459, 9434, 9471, 9519, 9556, 9948, 9960, 9965, 10033, 10045, 10050, 12044, 12046, 12063, 12129, 12131, 12148, 13119, 13136, 13204, 13221, 13634, 13635, 13651, 13652, 13719, 13720, 13736, 13737, 14136, 14170, 14221, 14255, 14665, 14695, 14699, 14750, 14780, 14784, 15178, 15185, 15219, 15263, 15270, 15304, 16231, 16240, 16248, 16316, 16325, 16333, 18342, 18362, 18427, 18447, 18868, 18873, 18953, 18958, 21993, 22078, 23027, 23112, 23552, 23590, 23637, 23675, 24076, 24103, 24161, 24188, 27216, 27219, 27301, 27304, 27730, 27742, 27764, 27815, 27827, 27849, 29826, 29860, 29911, 29945, 30884, 30895, 30912, 30918, 30969, 30980, 30997, 31003, 31935, 31952, 32020, 32037, 32447, 32460, 32464, 32532, 32545, 32549, 32977, 32984, 33001, 33062, 33069, 33086, 34022, 34047, 34056, 34107, 34132, 34141, 36127, 36141, 36212, 36226, 36667, 36752, 38719, 38734, 38751, 38753, 38804, 38819, 38836, 38838, 39769, 39854, 40290, 40291, 40324, 40325, 40375, 40376, 40409, 40410, 41334, 41419, 41885, 41970, 42930, 42947, 43015, 43032

Commentary

A = B ⊕ 85·{0,1} where B is the parabola in the group (ℤ/85)², placed in ℤ/44455 = ℤ/(85·523) by CRT with the second coordinate confined to a short interval. Since 85 = 5·17 and both 5, 17 ≡ 5 (mod 12), 3 is a quadratic non-residue mod each, so Ruzsa’s identity 3(x_a−x_b)² = 4(x_c−x_d)² forces triviality and B is exactly valid. Composite g gives sparser squares, hence a shorter v-support span, hence a smaller M.

last edited by Rayan Hatout at 2026-09-01 09:14:21 UTC · history