A sweetener problem in Number Theory for RMO

(I picked the following problem from Prof Titu Andreescu’s literature).

Problem:

Let p be a prime greater than 5. Prove that $p-4$ cannot be the fourth power of an integer.

Proof:

(By contradiction)

Assume that $p-4=q^{4}$ for some positive integer q. Then, $p=q^{4}+4$ and $q>1$ . We obtain

$p=q^{4}+4q^{2}+4-4q^{2}=(q^{2}+2)^{2}-(2q)^{2}=(q^{2}-2q+2)(q^{2}+2q+2)$ , which is a product of two integers greater than 1, contradicting the fact that p is a prime. (Note that for $p>5$ , $q>1$ , and so $(q-1)^{2}=q^{2}-2q+1>0$ , or $q^{2}-2q+2>1$ .