witness #1005

N: 22,475
|A|: 100
exponent log |A| / log N: 0.4596

status: current record for this modulus
submitted by: David Renshaw
submitted at: 2026-08-06 19:00:27 UTC

Elements (100)

0, 1, 5, 6, 28, 30, 33, 35, 86, 91, 1160, 1161, 1165, 1166, 1188, 1190, 1193, 1195, 1246, 1251, 2900, 2901, 2905, 2906, 2928, 2930, 2933, 2935, 2986, 2991, 4060, 4061, 4065, 4066, 4088, 4090, 4093, 4095, 4146, 4151, 5655, 5656, 5660, 5661, 5683, 5685, 5688, 5690, 5741, 5746, 8555, 8556, 8560, 8561, 8583, 8585, 8588, 8590, 8641, 8646, 8990, 8991, 8995, 8996, 9018, 9020, 9023, 9025, 9076, 9081, 9570, 9571, 9575, 9576, 9598, 9600, 9603, 9605, 9656, 9661, 11890, 11891, 11895, 11896, 11918, 11920, 11923, 11925, 11976, 11981, 12470, 12471, 12475, 12476, 12498, 12500, 12503, 12505, 12556, 12561

Commentary

Mixed digit lift across non-coprime factors: 22475 = 145·155, gcd = 5, so CRT does not apply — but the general lift B = {a + 145·b : a in A1, b in A2} works for any factorization because the coefficients of a + 3b − 2c − 2d sum to zero. Reducing a solution mod 145 forces the low parts to solve the equation mod 145, hence all equal (A1 is solution-free), making the low contribution vanish exactly as integers; the carry then forces the high parts to solve the equation mod 155, hence also equal. Here A1 is Bhavik Mehta's size-10 witness for 145 (score 0.8305) and A2 his size-10 witness for 155 (score 0.8032), giving |B| = 100 and score 100/√22475 ≈ 0.667 — the product of the factor scores. Verified locally by exhaustive pair-count convolution (exactly 100 solutions, all trivial).

last edited by David Renshaw at 2026-08-06 19:00:27 UTC · history