Computer-aided proof of Erdős discrepancy properties

Konev, Boris ORCID: 0000-0002-6507-0494 and Lisitsa, Alexei (2015) Computer-aided proof of Erdős discrepancy properties. Artificial Intelligence, 224. pp. 103-118.

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Abstract

In 1930s Paul Erdős conjectured that for any positive integer C in any infinite ±1 sequence (xn) there exists a subsequence xd, x2d, x3d, ..., xkd, for some positive integers k and d, such that |∑ki=1xi·d| >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 Erdős 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.

Item Type:	Article
Additional Information:	Revised and extended journal version of arXiv:1402.2184, http://arxiv.org/abs/1402.2184
Uncontrolled Keywords:	Erdős discrepancy problem, Computer-aided proof, Propositional satisfiability
Subjects:	?? QA75 ??
Depositing User:	Symplectic Admin
Date Deposited:	31 Mar 2016 10:09
Last Modified:	15 Dec 2022 13:37
DOI:	10.1016/j.artint.2015.03.004
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/2022487