Show that the pattern of prime gaps between the primes
5, 7, 11, 13, 17, 19 never repeats again.
Hint: consider n + i(modm) where i is in the given set and m is chosen carefully.
Quite a few questions in (medium)advanced number theory you've asked...wouldn't it be nice if you show us some selfwork? What have you done so far, where're you stuck...?
It also could be interesting to know from where do these question stem: mere curiosity, a course's exercises, research...?
Tonio