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Let x and n be positive integers such that $1 + x + x^2 + ... + x^{n-1}$ 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 = $1 + x + x^2 + ... + x^{n-1} = (1 + x + x^2 + ... + x^{a-1} ) + x^a (1 + x + x^2 + ... + x^{a-1} ) + x^{2a} (1 + x + x^2 + ... + x^{a-1} ) + ... + x^{a(b-1)} (1 + x + x^2 + ... + x^{a-1} )$

=$(1+x+x^2 +...+x^{a-1}) (x^a + x^{2a} +...+ x^{a(b-1)})$

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.