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Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer

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Theory and Applications of Satisfiability Testing – SAT 2016 (SAT 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9710))

Abstract

The boolean Pythagorean Triples problem has been a longstanding open problem in Ramsey Theory: Can the set \(\mathbb {N}= \{1,2,\dots \}\) of natural numbers be divided into two parts, such that no part contains a triple (a, b, c) with \(a^2 + b^2 = c^2\) ? A prize for the solution was offered by Ronald Graham over two decades ago. We solve this problem, proving in fact the impossibility, by using the Cube-and-Conquer paradigm, a hybrid SAT method for hard problems, employing both look-ahead and CDCL solvers. An important role is played by dedicated look-ahead heuristics, which indeed allowed to solve the problem on a cluster with 800 cores in about 2 days. Due to the general interest in this mathematical problem, our result requires a formal proof. Exploiting recent progress in unsatisfiability proofs of SAT solvers, we produced and verified a proof in the DRAT format, which is almost 200 terabytes in size. From this we extracted and made available a compressed certificate of 68 gigabytes, that allows anyone to reconstruct the DRAT proof for checking.

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Notes

  1. 1.

    In practice this is done using a single incremental CNF file. For details about the format, see http://www.siert.nl/icnf/.

  2. 2.

    Files and tools can be downloaded at http://www.cs.utexas.edu/~marijn/ptn/.

  3. 3.

    https://www.tacc.utexas.edu/systems/stampede.

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Acknowledgements

The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing grid resources that have contributed to the research results reported within this paper.

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Authors and Affiliations

  1. The University of Texas at Austin, Austin, USA

    Marijn J. H. Heule

  2. Swansea University, Swansea, UK

    Oliver Kullmann

  3. University of Kentucky, Lexington, USA

    Victor W. Marek

Authors
  1. Marijn J. H. Heule
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  2. Oliver Kullmann
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  3. Victor W. Marek
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Correspondence to Marijn J. H. Heule .

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Editors and Affiliations

  1. Aix-Marseille Université, Marseille, France

    Nadia Creignou

  2. Université d'Artois, Lens, France

    Daniel Le Berre

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Heule, M.J.H., Kullmann, O., Marek, V.W. (2016). Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer. In: Creignou, N., Le Berre, D. (eds) Theory and Applications of Satisfiability Testing – SAT 2016. SAT 2016. Lecture Notes in Computer Science(), vol 9710. Springer, Cham. https://doi.org/10.1007/978-3-319-40970-2_15

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  • DOI: https://doi.org/10.1007/978-3-319-40970-2_15

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