Square lift of the N=145 record. If A1 and A2 are both solution-free mod m, then B = {a + m*b : a in A1, b in A2} is solution-free mod m^2: reducing a putative solution mod m makes the low parts a solution mod m, so they are all equal, which makes the low contribution vanish exactly (the coefficients satisfy 1+3-2-2=0); the remaining carry then forces the high parts to be a solution mod m, hence also all equal. Taking A1 = A2 = Bhavik Mehta's size-10 witness for m=145 (score 0.8305) gives |B| = 100 at N = 145^2 = 21025, score 100/145 ≈ 0.6897. Greedy augmentation confirms the set is maximal: no single element of Z/21025 can be added without creating a nontrivial solution. Verified locally by exhaustive pair-count convolution (exactly |B| solutions, all trivial) before submission.
David Renshaw · 2026-08-06 18:56:36 UTC