With prime numbers $a < b < c$ and no primes exist in ranges $(a, b)$ and $(b, c)$, is it possible that there exists positive integers $x$, $y$, $z$ such that $|a^x c^z-b^y|=2$?
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1$\begingroup$ Do you intend to rule out the case a=2? Gerhard "Two Is Also A Prime" Paseman, 2020.01.31. $\endgroup$– Gerhard PasemanCommented Jan 31, 2020 at 15:15
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$\begingroup$ @GerhardPaseman If $a=2$, $|a^x c^z - b^y|$ is odd, so ... $\endgroup$– Robert IsraelCommented Jan 31, 2020 at 17:26
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$\begingroup$ So the one case where you can prove there are such terms close to each other is ruled out. Gerhard "Distance One Seems Not Wanted" Paseman, 2020.01.31. $\endgroup$– Gerhard PasemanCommented Jan 31, 2020 at 19:34
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$\begingroup$ Write a as b-alpha, c as b+beta, and delta as alpha times beta. Modulo delta, the sum is b^y(b^{x+z-y)-1) + (zbeta - xalpha)b^{y-1}. Since alpha and beta are both even, there are lots of times when this implies the difference is divisible by four. Gerhard "Not Seeing Many Solutions Here" Paseman, 2020.01.31. $\endgroup$– Gerhard PasemanCommented Feb 1, 2020 at 4:15
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