Does there exist three consecutive positive integers such that each of them is the power of a prime i.e., is there exist $n \in \mathbb{N}$, such that $n=p^i$, $n+1 = q^j$ and $n+2 = r^k$, where $p$, $q$ and $r$ are primes and $i,j,k >1$.
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Note that either both $n$ and $n+2$ are even, or both are odd. If both are even, then $p=r=2$ and we are done. If they are both odd, $n+1$ is even and $q=2$. So $$n+1=2^j\implies n=2^j-1=p^k$$ Now look at When is $2^n\pm1$ a perfect power.
chubakueno
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No. Catalan conjecture...................
Will Jagy
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1In short, for those who don't know what this is: the only instance of consecutive prime powers is the pair (8, 9). – Théophile May 29 '14 at 04:35
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2@Théophile Actually Catalan's conjecture (now a theorem) gives the stronger result that the only instance of consecutive integer powers (prime integers or not) is the pair $(8,9)$. – Ethan Splaver May 29 '14 at 04:45
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@Ethan Indeed, I see that you are right. Thank you for the clarification. – Théophile May 29 '14 at 04:49
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1Catalan is an overkill. $q$ needs to be $2$. – chubakueno May 29 '14 at 04:50