A sweetener problem in Number Theory for RMO

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


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


(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.

(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