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)


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.


  1. 1.
    Mizar proof checker. Accessed: November 2015Google Scholar
  2. 2.
    Coq proof manager. Accessed: November 2015Google Scholar
  3. 3.
    The site of \(flyspeck\) project, the formal verification of the proof of Kepler Conjecture. Accessed: November 2015Google Scholar
  4. 4.
    Ahmed, T., Kullmann, O., Snevily, H.: On the van der Waerden numbers w(2; 3, t). Discrete Appl. Math. 174, 27–51 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Biere, A.: Picosat essentials. JSAT 4(2–4), 75–97 (2008)MATHGoogle Scholar
  6. 6.
    Biere, A.: Lingeling, Plingeling and Treengeling entering the SAT competition 2013. In: Proceedings of SAT Competition 2013, p. 51 (2013)Google Scholar
  7. 7.
    Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press, Amsterdam (2009)MATHGoogle Scholar
  8. 8.
    Clarke, E.M., Grumberg, O., Peled, D.: Model Checking. MIT Press, Cambridge (1999)Google Scholar
  9. 9.
    Cook, S.A.: The complexity of theorem-proving procedures. In: Proceedings 3rd Annual ACM Symposium on Theory of Computing (STOC 1971), pp. 151–158 (1971)Google Scholar
  10. 10.
    Cooper, J., Overstreet, R.: Coloring so that no Pythagorean triple is monochromatic (2015). arXiv:1505.02222
  11. 11.
    Cooper, J., Poirel, C.: Note on the Pythagorean triple system (2008)Google Scholar
  12. 12.
    Crawford, J., Ginsberg, M., Luks, E., Roy, A.: Symmetry-breaking predicates for search problems. In: Proceedings of 5th International Conference on Knowledge Representation and Reasoning, KR 1996, pp. 148–159. Morgan Kaufmann (1996)Google Scholar
  13. 13.
    Dequen, G., Dubois, O.: kcnfs: an efficient solver for random k-SAT formulae. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 486–501. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Dransfield, M.R., Marek, V.W., Truszczyński, M.: Satisfiability and computing van der Waerden numbers. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 1–13. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  15. 15.
    Dubois, O., Dequen, G.: A backbone-search heuristic for efficient solving of hard 3-SAT formulae. In: International Joint Conferences on Artificial Intelligence (IJCAI), pp. 248–253 (2001)Google Scholar
  16. 16.
    Eén, N., Biere, A.: Effective preprocessing in SAT through variable and clause elimination. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 61–75. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Franco, J., Martin, J.: A history of satisfiability. In: Biere et al. [7], Chap. 1, pp. 3–74Google Scholar
  19. 19.
    Garey, M.R., Johnson, D.S.: Computers and Intractability/A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)MATHGoogle Scholar
  20. 20.
    Henschen, L.J., Wos, L.: Unit refutations and Horn sets. J. Assoc. Comput. Mach. 21(4), 590–605 (1974)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Heule, M.J.H., Biere, A.: Clausal proof compression. In: 11th International Workshop on the Implementation of Logics (2015)Google Scholar
  22. 22.
    Heule, M.J.H., Biere, A.: Compositional propositional proofs. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) LPAR-20 2015. LNCS, vol. 9450, pp. 444–459. Springer, Heidelberg (2015). doi:10.1007/978-3-662-48899-7_31 CrossRefGoogle Scholar
  23. 23.
    Heule, M.J.H., Hunt Jr., W.A., Wetzler, N.: Verifying refutations with extended resolution. In: Bonacina, M.P. (ed.) CADE 2013. LNCS, vol. 7898, pp. 345–359. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  24. 24.
    Heule, M.J.H., Hunt Jr., W.A., Wetzler, N.: Expressing symmetry breaking in DRAT proofs. In: Felty, A.P., Middeldorp, A. (eds.) CADE-25. LNCS, vol. 9195, pp. 591–606. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  25. 25.
    Heule, M.J.H., Kullmann, O., Wieringa, S., Biere, A.: Cube and conquer: guiding CDCL SAT solvers by lookaheads. In: Eder, K., Lourenço, J., Shehory, O. (eds.) HVC 2011. LNCS, vol. 7261, pp. 50–65. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  26. 26.
    Heule, M.J.H., van Maaren, H.: Look-ahead based SAT solvers. In: Biere et al. [7], Chap. 5, pp. 155–184Google Scholar
  27. 27.
    Järvisalo, M., Biere, A., Heule, M.J.H.: Blocked clause elimination. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 129–144. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  28. 28.
    Järvisalo, M., Heule, M.J.H., Biere, A.: Inprocessing rules. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 355–370. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  29. 29.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatche, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)CrossRefGoogle Scholar
  30. 30.
    Kay, W.: An overview of the constructive local lemma. Master’s thesis, University of South Carolina (2009)Google Scholar
  31. 31.
    Kouril, M.: Computing the van der Waerden number \({W}(3,4) = 293\). INTEGERS: Electron. J. Comb. Number Theory 12(A46), 1–13 (2012)Google Scholar
  32. 32.
    Kouril, M., Paul, J.L.: The van der Waerden number \({W}(2,6)\) is \(1132\). Exp. Math. 17(1), 53–61 (2008)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Kullmann, O.: On a generalization of extended resolution. Discrete Appl. Math. 96–97, 149–176 (1999)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Kullmann, O.: Fundaments of branching heuristics. In: Biere et al. [7], Chap. 7, pp. 205–244Google Scholar
  35. 35.
    Levin, L.: Universal search problems. Problemy Peredachi Informatsii 9, 115–116 (1973)MATHGoogle Scholar
  36. 36.
    Li, C.M.: A constraint-based approach to narrow search trees for satisfiability. Inf. Process. Lett. 71(2), 75–80 (1999)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Li, C.M., Anbulagan: Heuristics based on unit propagation for satisfiability problems. In: Proceedings of 15th International Joint Conference on Artificial Intelligence (IJCAI 1997), pp. 366–371. Morgan Kaufmann Publishers (1997)Google Scholar
  38. 38.
    Manthey, N., Heule, M.J.H., Biere, A.: Automated reencoding of boolean formulas. In: Proceedings of Haifa Verification Conference 2012 (2012)Google Scholar
  39. 39.
    Marques-Silva, J.P., Lynce, I., Malik, S.: Conflict-driven clause learning SAT solvers. In: Biere et al. [7], Chap. 4, pp. 131–153Google Scholar
  40. 40.
    Mijnders, S., de Wilde, B., Heule, M.J.H.: Symbiosis of search and heuristics for random 3-SAT. In: Mitchell, D., Ternovska, E. (eds.) Third International Workshop on Logic and Search (LaSh 2010) (2010)Google Scholar
  41. 41.
    Myers, K.J.: Computational advances in Rado numbers. Ph.D. thesis, Rutgers University (2015)Google Scholar
  42. 42.
    Rado, R.: Some partition theorems. In: Colloquia Mathematica Societatis János Bolyai 4. Combinatorial Theory and Its Applications III, pp. 929–936. North-Holland, Amsterdam (1970)Google Scholar
  43. 43.
    Schur, I.: Über die Kongruenz \(x^m + y^m = z^m\) (mod p). Jahresbericht der Deutschen Mathematikervereinigung 25, 114–117 (1917)Google Scholar
  44. 44.
    Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Siekmann, J.H., Wrightson, G. (eds.) Automation of Reasoning 2, pp. 466–483. Springer, Heidelberg (1983)CrossRefGoogle Scholar
  45. 45.
    van der Waerden, B.L.: Beweis einer Baudetschen Vermutung. Nieuw Archief voor Wiskunde 15, 212–216 (1927)MATHGoogle Scholar
  46. 46.
    Van Gelder, A.: Verifying RUP proofs of propositional unsatisfiability. In: ISAIM (2008)Google Scholar
  47. 47.
    Voevodski, V.: Lecture at ASC 2008, How I became interested in foundations of mathematics. Accessed: November 2015Google Scholar
  48. 48.
    Wetzler, N., Heule, M.J.H., Hunt Jr., W.A.: DRAT-trim: efficient checking and trimming using expressive clausal proofs. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 422–429. Springer, Heidelberg (2014)Google Scholar
  49. 49.
    Zhang, H.: Combinatorial designs by SAT solvers. In: Biere et al. [7], Chap. 17, pp. 533–568Google Scholar
  50. 50.
    Zhang, L., Malik, S.: Validating SAT solvers using an independent resolution-based checker: practical implementations and other applications. In: DATE, pp. 10880–10885 (2003)Google Scholar

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

Personalised recommendations