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

  title={Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer},
  author={Marijn J. H. Heule and Oliver Kullmann and Victor W. Marek},
The boolean Pythagorean Triples problem has been a longstanding open problem in Ramsey Theory: Can the set N = $\{1, 2, ...\}$ 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… 
Formally Proving the Boolean Pythagorean Triples Conjecture
This work formalizes the Boolean Pythagorean Triples problem in Coq, recursively defining a family of propositional formulas, parameterized on a natural number n, and showing that unsatisfiability of this formula for any particular n implies that there does not exist a solution to the problem.
Cube-and-Conquer for Satisfiability
This chapter presents the cube-and-conquer paradigm, a competitive alternative for solving SAT problems in parallel which outperforms both lookahead and conflict-driven solvers and achieves linear-time speedups on this problem even when using thousands of cores.
Formally Verifying the Solution to the Boolean Pythagorean Triples Problem
This work state the Boolean Pythagorean Triples problem in Coq, define its encoding as a propositional formula and establish the relation between solutions to the problem and satisfying assignments to the formula.
Everything's Bigger in Texas: "The Largest Math Proof Ever"
The Boolean Pythagorean triples problem is answered by encoding it into propositional logic and applying massive parallel SAT solving on the resulting formula, and a proof of the solution is produced, called the “largest math proof ever”.
Schur Number Five
A proof of the solution of a century-old problem, known as Schur Number Five, is constructed and validated using a formally verified proof checker, demonstrating that any result by satisfiability solvers can now be validated using highly trustworthy systems.
A SAT + CAS Verifier for Combinatorial Conjectures
Using MATHCHECK2, a combination of a SAT solver and a computer algebra system aimed at finitely verifying or counterexampling mathematical conjectures, the Hadamard conjecture was verified from design theory and independent verification of the claim that Williamson matrices of order 35 do not exist is provided.
Investigating the Existence of Large Sets of Idempotent Quasigroups via Satisfiability Testing
A method is described for solving some open problems in design theory based on SAT solvers by using an incremental search strategy to find a maximum number of disjoint idempotent quasigroups, thus deciding the non-existence of large sets.
Combining SAT Solvers with Computer Algebra Systems to Verify Combinatorial Conjectures
This paper presents a method and an associated system, called MathCheck, that embeds the functionality of a computer algebra system (CAS) within the inner loop of a conflict-driven clause-learning SAT solver, and leverages the capabilities of several different CAS, namely the SAGE, Maple, and Magma systems.
The SAT+CAS method for combinatorial search with applications to best matrices
An overview of the SAT+CAS method that combines satisfiability checkers (SAT solvers) and computer algebra systems (CAS) to resolve combinatorial conjectures, and present new results vis-à-vis best matrices.
Proof Search and the Structure of Solution Spaces
In 2019, Moritz Müller and I resolved a longstanding open question on the computational complexity of proof search. The preliminary version of the article “Automating Resolution is NP-Hard” was the


Satisfiability and Computing van der Waerden Numbers
This paper shows that by using general-purpose complete and local-search techniques for testing propositional satisfiability, this approach becomes effective — competitive with specialized approaches.
Coloring so that no Pythagorean Triple is Monochromatic
It is proved that the hypergraph of Pythagorean triples can contain no Steiner triple systems, a natural obstruction to 2-colorability, and a SAT solver is used to find a 2-coloring for {1,...,7664}.
A backbone-search heuristic for efficient solving of hard 3-SAT formulae
A heuristic search for variables belonging to the backbone of a 3-SAT formula which are chosen as branch nodes for the tree developed by a DPL-type procedure is defined, making it possible to handle unsatisfiable hard 3- SAT formulae up to 700 variables.
Cube and Conquer: Guiding CDCL SAT Solvers by Lookaheads
This work presents a new approach, called cube-and-conquer, targeted at reducing solving time on hard instances, and finds that this hybrid approach outperforms both lookahead and conflict-driven solvers.
Note on the Pythagorean Triple System
We investigate some combinatorial aspects of the “Pythagorean triple system”. Our motivation is the following question: Is it possible to color the naturals with finitely many colors so that no
The van der Waerden Number W(2, 6) Is 1132
The exhaustive search showing that W(2, 6) = 1132 was carried out by formulating the problem as a satisfiability (SAT) question for a Boolean formula in conjunctive normal form (CNF) and then using a SAT solver specifically designed for the problem.
kcnfs: An Efficient Solver for Random k-SAT Formulae
Experimental results show that kcnfs outperforms by far the best current complete solvers on any random k-SAT formula for k ≥ 3, and introduces a sub-heuristic called a re-normalization heuristic in order to handle formulae with various clause lengths.
Computers and Intractability: A Guide to the Theory of NP-Completeness
Horn formulae play a prominent role in artificial intelligence and logic programming. In this paper we investigate the problem of optimal compression of propositional Horn production rule knowledge
Handbook of Satisfiability: Volume 185 Frontiers in Artificial Intelligence and Applications
This collection of papers on all theoretical and practical aspects of SAT solving will be extremely useful to both students and researchers and will lead to many further advances in the field.
Compositional Propositional Proofs
A new framework to produce clausal proofs for cube-and-conquer, arguably the most effective parallel SAT solving paradigm for hard-combinatorial problems, is presented and an elegant approach to parallelize the validation of clausal Proofs efficiently is provided.