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


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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    $\begingroup$ Look up cyclotomic polynomials. In general, n+1 has to be prime also. These are part of the study of multiperfect numbers. They also appear in some accounts of certain Diophantine equations such as Catalan and Fermat. Gerhard "Check Ribenboim Books For More" Paseman, 2018.08.19. $\endgroup$ – Gerhard Paseman Aug 19 '18 at 20:42
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    $\begingroup$ The classification of doubly transitive permutation groups depends to some extent on knowing whether such ratios are prime or not (that is, given the classification of finite simple groups, doubly transitive permutation groups are known up to arithmetical questions similar to yours). So the primality (or otherwise) of such ratios is well-studied, but open questions remain. $\endgroup$ – Geoff Robinson Aug 19 '18 at 20:57
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    $\begingroup$ These are commonly referred to as "repunits" or "repdigits", in the setting where $p$ is replaced by an arbitrary base (but usually base 10). $\endgroup$ – Pace Nielsen Aug 19 '18 at 22:03
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    $\begingroup$ You'll find tables for $p=2,3,5,7,11$ at homes.cerias.purdue.edu/~ssw/cun/pmain518 $\endgroup$ – Gerry Myerson Aug 19 '18 at 23:19
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    $\begingroup$ There seems no special reason to restrict to prime values of p. $\endgroup$ – Aaron Meyerowitz Aug 20 '18 at 0:09

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