Question

Find all positive integers n for which \[\sqrt (n-1)+\sqrt (n+1)\] is rational

Answer 1

hmm
i think, there is no positive integer n which works...

Answer 2

Well, that may be true, but how can I prove that?

Answer 3

well
i was never good in proving things
but to get a rational solution sqr.(n-1) and sqr. (n+1) have to be rational, otherwise a rational number would be added to a irrational number.
so you have to find solutions for sqr. (n-1) which fit into the given conditions like:
M = {2; 5; 10...}
and the answers for sqr. (n+1):
N= {0, 3, 8...}

both sets have no common numbers and will never have one so there is no n which solves the task.

that is no proof, but maybe it will help you or show that i got it wrong xD