(I picked the following problem from Prof Titu Andreescu’s literature).
Problem:
Let p be a prime greater than 5. Prove that cannot be the fourth power of an integer.
Proof:
(By contradiction)
Assume that for some positive integer q. Then,
and
. We obtain
, which is a product of two integers greater than 1, contradicting the fact that p is a prime. (Note that for
,
, and so
, or
.
(Ref: 104 Number Theory Problems From the Training of the USA IMO Team by Titu Andreescu, Dorin Andrica, Zumin Feng).
More sweeteners in number theory coming soon,
Nalin Pithwa