Let x and n be positive integers such that is a prime number. Then show that n is a prime number.

Solution:

(For small values of x and n it is easy to show that the given fact is true. We prove for x>>1)

Suppose n is not a prime. Then n = ab (where both a and b are not equal to 1). We may write the given expression in blocks of a terms; there will be b such blocks.

prime =

=

But this gives a factorization of a prime number which is not possible (as x>>1 none of the factor equals 1). Hence we find a contradiction implying n is prime.