Computer-aided proof of Erdős discrepancy properties



(2015) Computer-aided proof of Erdős discrepancy properties. Artificial Intelligence, 224. pp. 103-118. ISSN 1872-7921 (Online); 0004-3702 (Print)

[img] Text
journal.pdf

Download (354kB)

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 | xd + x2d + ... + xid | > 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
Subjects: Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Depositing User: Symplectic Admin
Date Deposited: 31 Mar 2016 10:09
Last Modified: 31 Oct 2018 10:07
DOI: 10.1016/j.artint.2015.03.004
URI: http://livrepository.liverpool.ac.uk/id/eprint/2022487
Repository Staff Access