witness #234

N: 49,999
|A|: 114
score |A|/√N: 0.5098
exponent log |A| / log N: 0.4377

status: current record for this modulus
submitted by: David Renshaw
submitted at: 2026-08-04 01:52:24

Elements (114)

0, 1, 5, 14, 18, 19, 160, 161, 165, 174, 178, 179, 800, 801, 805, 814, 818, 819, 960, 961, 965, 974, 978, 979, 2000, 2001, 2005, 2014, 2018, 2019, 2160, 2161, 2165, 2174, 2178, 2179, 2800, 2801, 2805, 2814, 2818, 2819, 2960, 2961, 2965, 2974, 2978, 2979, 10000, 10001, 10005, 10014, 10018, 10019, 10800, 10801, 10805, 10814, 10818, 10819, 12000, 12001, 12005, 12014, 12018, 12019, 12800, 12801, 12805, 12814, 12818, 12819, 20160, 20161, 20165, 20174, 20178, 20179, 20480, 20481, 20485, 20494, 20498, 20499, 20960, 20961, 20965, 20974, 20978, 20979, 21280, 21281, 21285, 21294, 21298, 21299, 22160, 22161, 22165, 22174, 22178, 22179, 22480, 22481, 22485, 22494, 22498, 22499, 22960, 22961, 22965, 22974, 22978, 22979

Commentary

Found by transplanting witness#229 for N = 50,000 unchanged to N = 49,999. Its digit-product structure explains why this works: A = L + 80K, where L = {0, 1, 5, 14, 18, 19} and K is witness#106 modulo 625. Write an integer discrepancy as D = D_L + 80D_K. Since |D| < 2 · 49,999, a new relation modulo 49,999 would require D = ±49,999. The bound |D_L| ≤ 76 then forces (D_L, D_K) = (−1, 625) or (1, −625), contradicting the validity of K modulo 625. The case D = 0 was already ruled out by witness#229.

last edited by David Renshaw at 2026-08-04 02:22:56 · history