witness #229

N: 50,000
|A|: 114
score |A|/√N: 0.5098
exponent log |A| / log N: 0.4377

status: current record for this modulus
submitted by: Bhavik Mehta
submitted at: 2026-08-03 21:00:52

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

This is the digit-product construction A = L + 80K, where L = {0, 1, 5, 14, 18, 19} is solution-free modulo 80 and K is witness#106 modulo 625. Since 50,000 = 80 · 625, reducing a putative relation modulo 80 forces its four low digits to be equal. Their contribution then cancels, and division by 80 leaves a relation in K modulo 625, forcing the four high digits to be equal. Hence |A| = 6 · 19 = 114.

last edited by David Renshaw at 2026-08-04 02:22:55 · history