Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Büchi's 5-square problem by Joseph LIPMAN, Purdue University Abstract: A Büchi n-tuple x1 < x2 < ... < xn of nonnegative integers is a solution of the system x12 - 2x22 + x32 = x22 - 2x32 + x42 = ... = xn-22 - 2xn-12 + xn2 = 2, not of the `trivial' form (m,m+1, ... , m+n-1), with m Î N. The case n=3 is elementary, and there are infinitely many Büchi 4-tuples (can you find one?), but no one knows whether or not Büchi 5-tuples exist. Büchi raised this question in the 1970s after noting that a negative answer---even with 5 replaced by some larger n---would lead from Matiyesevich's theorem to nonexistence of an algorithm for deciding whether any diagonal diophantine system å aij xj2 = bi has a solution. Work of Vojta on connections with a conjecture of Lang on surfaces of general type will be touched on, as well as an unproved conjecture on Büchi 4-tuples which holds for x2 < 10500.