Speaker: Stanley Yao Xiao
Abstract: In the aftermath of Matiyasevich’s negative solution to Hilbert’s Tenth Problem, J.R. Buchi gave an argument showing that even much simpler systems, those defined by an arbitrary number of diagonal quadratic forms in an arbitrary number of indeterminates, cannot be decided over the integers. His argument depended on an unproven hypothesis which is now known as Buchi’s problem: there exists a positive integer n₀ such that whenever n ≥ n₀ and x₁², …, x_n² are n increasing integer squares with constant second difference equal to 2, they must be consecutive. Buchi suggested that it suffices to take n₀ = 5. We prove this assertion, thereby completing Buchi’s argument that arbitrary systems of diagonal quadratic form equations are undecidable over the integers.