I recently saw a question in MSE asking if there are any relatively prime integers $a\not=1,b\not=0$ and such that $a^k+b$ will hit infinitely many primes, and the answer was that it's an open problem even for basic cases as $a=2,3$.

So my question is : Is it known that for some $a,b$ it will never hit a prime?

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    $\begingroup$ Sure. Just let $a=b$. Then $a^k+b=a(a^{k-1}-1)$ and it cannot be prime as long as neither $a$ or $a-1$ are prime. $\endgroup$ – Franklin Pezzuti Dyer May 29 '17 at 21:59
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    $\begingroup$ Oh right. Then let $a$ and $b$ both be odd and relatively prime, and both not equal to one. Then $a^k+b$ is even and greater than two, and thus not prime. $\endgroup$ – Franklin Pezzuti Dyer May 29 '17 at 22:12
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    $\begingroup$ If there is a prime $p$ such that $a \equiv 1 \pmod{p}$ and $b \equiv -1 \pmod{p}$, then $a^k + b$ will always be a multiple of $p$. We can also find examples where there is no prime dividing all the $a^k + b$. $\endgroup$ – Daniel Fischer May 29 '17 at 22:48
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    $\begingroup$ Just to be clear, I mainy posted my $0^k+1$ comment as an instructive joke. The "instructive" part is that it's always a good idea, when trying to formulate a problem precisely, to check for trivial answers. The OP had actually done this to a large extent in ruling out $a=1$ and/or $b=0$. $\endgroup$ – Barry Cipra May 29 '17 at 23:06
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    $\begingroup$ For example, for $a = 32$ and $b = 358423$ the number $a^k + b$ is never prime, and no prime divides all of these numbers. $\endgroup$ – Daniel Fischer May 29 '17 at 23:09

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