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# Prime as a sum of Geometric Series

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.