Computer-Aided Proof of Erdos Discrepancy Properties

In 1930s Paul Erdos conjectured that for any positive integer C in any infinite ± 1 sequence (xn) there exists a subsequence xd, x2d, x3d,dots, xkd, for some positive integers k and d, such that mid sumi=1k xi· d mid >C. The conjecture has been referred to as one of the major open problems in combinatorial number theory and discrepancy theory. For the particular case of C=1 a human proof of the conjecture exists; for C=2 a bespoke computer program had generated sequences of length 1124 of discrepancy 2, but the status of the conjecture remained open even for such a small bound. We show that by encoding the problem into Boolean satisfiability and applying the state of the art SAT solvers, one can obtain a discrepancy 2 sequence of length 1160 and a proof of the ErdHos discrepancy conjecture for C=2, claiming that no discrepancy 2 sequence of length 1161, or more, exists. In the similar way, we obtain a precise bound of 127,645 on the maximal lengths of both multiplicative and completely multiplicative sequences of discrepancy 3. We also demonstrate that unrestricted discrepancy 3 sequences can be longer than 130,000.