Everything's Bigger in Texas: "The Largest Math Proof Ever"

Abstract

Progress in satisfiability (SAT) solving has enabled answering long-standing open questions in mathematics completely automatically resulting in clever though potentially gigantic proofs. We illustrate the success of this approach by presenting the solution of the Boolean Pythagorean triples problem [7]. We also produced and validated a proof of the solution, which has been called the "largest math proof ever" [12]. The enormous size of the proof is not important. In fact a shorter proof would have been preferable. However, the size shows that automated tools combined with super computing facilitate solving bigger problems. Moreover, the proof of 200 terabytes can now be validated using highly trusted systems [5,3,13], demonstrating that we can check the correctness of proofs no matter their size.The We answer this question, known as the Boolean Pythagorean triples problem, by encoding it into propositional logic and applying massive parallel SAT solving on the resulting formula. More concretely, we search for the smallest number n such that every coloring of the numbers 1 to n with red and blue results in a monochromatic solution of a 2 + b 2 = c 2 . For each number i a Boolean variable v i is introduced. If v i is assigned to true (or false), then number i is colored red (or blue). For each solution of a 2 + b 2 = c 2 , the propositional formula contains a clause stating that at least one of a, b, and c must be colored red (v a ∨ v b ∨ v c ) and at least one of a, b, and c must be colored blue (v a ∨ v b ∨ v c ). This formula is simplified, by removing redundant Pythagorean triples and symmetry breaking, before solving it. * Thanks to the co-authors of the work summarized here: