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A SAT Attack on the Erdős Discrepancy Conjecture

  • Boris Konev
  • Alexei Lisitsa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8561)

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.

Keywords

Discrepancy Theory Arithmetic Progression Boolean Formula Satisfying Assignment Bound Model Check 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Boris Konev
    • 1
  • Alexei Lisitsa
    • 1
  1. 1.Department of Computer ScienceUniversity of LiverpoolUnited Kingdom

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