Stanley Xiao, University of Northern British Columbia
On Buchi's problem
In 1970, J. Richard Buchi showed that there is no general algorithm which decides whether a general quadratic equation in arbitrarily many variables has a solution in the integers, subject to a hypothesis which would be named Buchi's Problem. Buchi's result is a strengthening of the negative answer of Hilbert's Tenth Problem.
Buchi's problem is an elegant number theoretic problem in its own right. It asserts that there exists a positive integer M such that whenever a finite sequence x_0^2, x_1^2,..., x_n^2 of increasing square integers has constant second difference equal to 2 (that is, x_{j+2}^2 - 2 x_{j+1}^2 + x_j^2 = 2 for j = 0, ..., n-2), then either n \leq M or x_j^2 = (x_0 + j)^2 for j = 1, ..., n.
In this talk, we show that Buchi's problem has an affirmative answer with M = 5. In other words, there are no non-trivial quintuple of increasing square integers with constant difference equal to 2.
MC 5479