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

  • Marijn J. H. Heule
  • Oliver Kullmann
  • Victor W. Marek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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 (abc) 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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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Marijn J. H. Heule
    • 1
  • Oliver Kullmann
    • 2
  • Victor W. Marek
    • 3
  1. 1.The University of Texas at AustinAustinUSA
  2. 2.Swansea UniversitySwanseaUK
  3. 3.University of KentuckyLexingtonUSA

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