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Consider the sequence of numbers ranging over $n$ of the form $k\cdot a^n + 1$ for a fixed, odd natural number $k$. The number $k$ is considered a Sierpinski number if the sequence determined by that $k$ contains no primes.

One way to show that such a $k$ is a Sierpinski number is to find a covering system of congruences for the sequence proving there are no primes in the sequence. The way to show that such a $k$ is not a Sierpinski number is to find one prime among the numbers in the sequence.

For those sequences where a prime has been found, that is, $k$ is not a Sierpinski number, is it known whether there could be infinitely many primes in that sequence?

If only a finite number of primes are expected to be in the sequence, then a covering system of congruences might work after a certain $n$ to show that the rest of the sequence contains only composite numbers.

Sierpinski showed that there are an infinite number of $k$ whose sequences contain no primes. My conjecture is that there can be at most a finite number of primes for any $k$, but I wonder if this has not already been resolved by someone or if someone has already made the conjecture.

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    $\begingroup$ This might be worth cross-posting in MathOverflow if it doesn't get a quick response here. $\endgroup$ – Justin Benfield Mar 14 '16 at 15:10
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    $\begingroup$ Since we don't know whether there are even infinitely many Mersenne primes (of the form $2^n-1$, close enough), I doubt that this is known (alternatively, Fermat primes, of the form $2^n+1$, so $k=1$, also unknown whether or not there are infinitely many). $\endgroup$ – vrugtehagel Mar 14 '16 at 16:00
  • $\begingroup$ @Justin I'll see what response I get here and then cross-post it next week. Thanks for the suggestion. $\endgroup$ – Frank Hubeny Mar 14 '16 at 18:55
  • $\begingroup$ @vrugtehagel The Fermat primes would be a specific example of one of these sequences as you mention. I wonder if anyone has formulated a conjecture that all such sequences contain no more than a finite number of primes. I also wonder if anyone has found a $k$ for which a prime exists, but no more than finitely many primes can exist in the sequence. $\endgroup$ – Frank Hubeny Mar 14 '16 at 18:59

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