witness #1008

N: 49,735
|A|: 140
exponent log |A| / log N: 0.4569

status: current record for this modulus
submitted by: David Renshaw
submitted at: 2026-08-06 19:31:06 UTC

Elements (140)

0, 318, 381, 465, 758, 760, 1101, 2058, 2180, 2611, 3075, 4006, 4090, 4238, 4350, 4581, 4790, 4876, 4960, 5805, 5806, 6090, 6125, 6408, 6555, 6675, 6848, 7191, 7546, 7863, 8148, 8206, 8270, 8733, 8990, 9076, 10180, 10241, 10328, 10671, 11048, 11050, 11895, 11896, 12215, 12470, 12765, 13636, 13953, 14296, 14528, 14823, 15080, 15166, 16095, 16096, 16331, 16415, 16536, 16965, 17138, 17836, 18130, 18153, 18560, 19023, 20016, 20188, 20531, 20618, 22185, 22186, 22505, 22626, 23055, 23641, 23926, 24220, 24243, 24511, 25113, 25700, 26106, 26278, 26621, 26826, 27440, 28420, 29731, 29816, 30306, 30478, 30601, 31556, 31790, 32916, 33530, 33931, 34510, 34801, 35906, 35966, 35990, 36396, 36430, 36568, 37646, 37705, 37730, 39445, 39763, 39910, 40021, 40106, 40546, 40891, 41503, 41846, 42056, 42080, 42520, 43535, 43795, 43820, 44026, 44405, 45535, 45853, 46000, 46196, 46256, 46636, 46720, 47593, 47651, 47936, 47995, 48435, 48521, 49625

Commentary

CRT product construction: 49735 = 145 × 343 (coprime), and if A₁ ⊂ Z/N₁ and A₂ ⊂ Z/N₂ are both solution-free for a + 3b ≡ 2c + 2d, then the CRT image of A₁ × A₂ in Z/(N₁N₂) is solution-free too — a nontrivial solution would project to a solution in each factor that is nontrivial in at least one of them. This set is the product of the existing records for N=145 (size 10, witness 19) and N=343 (size 14, witness 94), both due to Bhavik Mehta, giving 10 × 14 = 140 elements. A greedy pass confirmed the set is maximal: no single element of Z/49735 can be added without creating a solution. Found and verified with a small C program maintaining exact ordered-solution counts (a set is valid iff the count equals |A|).

last edited by David Renshaw at 2026-08-06 19:31:06 UTC · history