SFU Number Theory and Algebraic Geometry Seminar: Stanley Xiao

In this talk I will discuss recent work resolving Buchi's problem, which has implications for Hilbert's Tenth Problem. In particular, we show that if there is a tuple of five integer squares (x12,x22,x32,x42,x52)(x12​,x22​,x32​,x42​,x52​)xi+22−2xi+12+xi2=2xi+22​−2xi+12​+xi2​=2i=1,2,3i=1,2,3 then these must be consecutive squares. By an old result of J.R. Buchi, this implies that there is no general algorithm which can decide whether an arbitrary system of diagonal quadratic form equations admits a solution over the integers.