On Hilbert's Tenth Problem and a conjecture of Buchi

Abstract: 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 $(x_1^2, x_2^2, x_3^2, x_4^2, x_5^2)$ satisfying $x_{i+2}^2 - 2x_{i+1}^2 + x_i^2 = 2$ for $i = 1,2,3$, 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.

algebraic geometrynumber theory

Audience: researchers in the discipline

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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