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.
David Renshaw · 2026-08-04 01:56:02
Found by transplanting witness#229 for N = 50,000 unchanged to N = 49,999.
David Renshaw · 2026-08-04 01:55:02
Found by transplanting witness#229 for N = 50,000 unchanged to N = 49,999. I downloaded the public witness database and tested the strongest nearby constructions at the target modulus. For this 114-element set, an exhaustive pair-sum check compared every value a + 3b with every value 2c + 2d modulo 49,999 and found only the permitted intersections a = b = c = d. Thus the same structured set remains valid after changing the modulus by one, improving the previous record from 69 to 114.
David Renshaw · 2026-08-04 02:22:56
David Renshaw · 2026-08-04 01:56:02
David Renshaw · 2026-08-04 01:55:02