poers of 2,3,5

prove or disprove: does there exist a positive integer N such that there exist integers r, s, t where N = r^2 + 1 = s^3 + 1 = t^5 +1. If so find the smallest such N and prove it is minimal.

*spoily?* * * * * * * * * * * * * * * * * * How about 1 = 0^2 + 1 = 0^3 + 1 = 0^5 + 1 ? Or 2 = 1^2 + 1 = ... The next one would be N = 2^30+1.

cheater how about some non trivial soutions

Well, the problem itself is pretty trivial. You want N-1 to be a square, a cube and a fifth power. Do you see why it must be a 30th power? Will any 30th power do?

More interesting might be something like: find the least positive k such that there exist a square, a cube and a fifth power which are all distinct, all greater than 1, and no two differ by more than k.

Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2

writes

N=1, r=s=t=0.

A couple of easy trivial solutions:

r = s = t = 0, N = 1 r = s = t = 1, N = 2

Next solution:

r = 2^15 s = 2^10 t = 2^6 N = 2^30 + 1