I don't know a non-computational solution to this question, or even a computational solution that wouldn't make one annoyed not to be allowed to use a calculator or computer. I don't know of any theoretical reason why the statement is false, and there are similar questions involving Euclid sequences that remain open: see e.g. Problem 6 of
http://alpha.math.uga.edu/~pete/NT2009HW1.pdf
A "U" in front of a part of the problem means that it is unsolved.
I am a professional mathematician specializing in number theory who has thought at least a little bit about similar problems in the context of teaching advanced undergraduate number theory. (I am not the world's greatest problem solver, but some of the people who have already commented on this question and not given a solution meeting the criteria above are about as quick and clever as they come.) So I submit that if I cannot solve the question in a nice way, then it is not reasonable to be able to expect an undergraduate to do so on an in-class exam.
