(I picked the following problem from Prof Titu Andreescu’s literature).
Let p be a prime greater than 5. Prove that cannot be the fourth power of an integer.
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,