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)


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.


Discrepancy Theory Arithmetic Progression Boolean Formula Satisfying Assignment Bound Model Check 
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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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