Is there an easier prime factorization method for the sum of a prime's powers?

Mathematics Asked by Ifn47 on August 3, 2020

I need to obtain prime factorizations of numbers of the type: $$sum_{i=0}^n p^i$$, for any prime number $$p$$ (not the same one each time).

Do you know if there is a quicker algorithm to calculate these factorizations than those used for other natural numbers?

I don’t know if there is a known solution. My only lead is that all Mersenne primes are of the form $$sum_{i=0}^n 2^i$$.

Edit: by prime factorization I mean, for example, if $$p$$ is 3 and $$n$$ is 6, the number is 364, and the prime factorization I’m looking for is 2^2, 7 and 13.

You have a geometric series, so $$sum_{i=0}^n p^i=frac {p^{n+1}-1}{p-1}$$. When the prime is $$2$$ this does not give a factorization because the denominator is $$1$$. For all other primes it does.

Correct answer by Ross Millikan on August 3, 2020

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