I am looking at prime numbers of the form $Q=1+p+p^2+\dots+p^n$, where $p$ is also a prime number, $n$ is finite. Unfortunately I was unable to find any reference to such prime numbers in the literature. Since this is a geometric series, we can obviously write it as

$$Q_p(k)=\frac{p^{k}-1}{p-1}$$

and a simple example are Mersenne prime numbers, i.e., $$M(n)=2^n-1.$$

Also obviously for $k=2$, there are no other such prime numbers except for the pair $(p=2, Q=3).$

An example for $p=3$, i.e., $Q_3(k)=\frac{3^k-1}{2}$ was discussed here prime-numbers-of-the-form-3n-1-2

Do these prime numbers have a name, are some general properties (e.g. their infinitude) known? Thanks.

Update:
I found that e.g. here http://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/home.html they are called *repunit primes*.

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