A SAT Attack on the Erdős Discrepancy Conjecture

Abstract

In 1930s Paul Erdős conjectured that for any positive integer C in any infinite ±1 sequence (x _n ) there exists a subsequence x _d , x _2d , x _3d ,…, x _kd , for some positive integers k and d, such that $\mid \sum_{i=1}^k x_{id} \mid >C$. 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 solver, one can obtain a discrepancy 2 sequence of length 1160 and a proof of the Erdős discrepancy conjecture for C = 2, claiming that no discrepancy 2 sequence of length 1161, or more, exists. We also present our partial results for the case of C = 3.