A SAT Attack on the Erdos Discrepancy Conjecture
Abstract
In 1930s Paul Erdos conjectured that for any positive integer C in any infinite +1 -1 sequence (x_n) there exists a subsequence x_d, x_{2d}, ... , x_{kd} for some positive integers k and d, such that |x_d + x_{2d} + ... + x_{kd}|> 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 having 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 sequence of length 1160 with discrepancy 2 and a proof of the Erdos discrepancy conjecture for C=2, claiming that no sequence of length 1161 and discrepancy 2 exists. We also present our partial results for the case of C=3.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2014
- DOI:
- 10.48550/arXiv.1402.2184
- arXiv:
- arXiv:1402.2184
- Bibcode:
- 2014arXiv1402.2184K
- Keywords:
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- Computer Science - Discrete Mathematics;
- Mathematics - Combinatorics;
- Mathematics - Number Theory;
- F.2.2;
- I.2.3;
- G.2.1
- E-Print:
- 8 pages. The description of the automata is clarified