3 consecutive numbers being (nontrivial) powers

rex88 · P versus NP Offline Joined: 12 Sep 2009 Posts: 22

are there any 3 consecutive numbers being (nontrivial) powers ?

( $n$ is nontrivial power iff there are $k,m>1$ such that $n=k^m$ )

there are two: $8=2^3$ and $9=3^2$

**Posted:** Mon Oct 19, 2009 12:37 pm

mavropnevma

Caaaaaaaaaataaaaaaaaaalaaaaaaaaaan!

Search Catalan's conjecture (now a theorem) on Google.

**Posted:** Mon Oct 19, 2009 1:29 pm

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ZetaX

Congratulations for making a trivial problem as hard as Mihailescu's theorem.

**Posted:** Mon Oct 19, 2009 1:57 pm

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rex88 · P versus NP Offline Joined: 12 Sep 2009 Posts: 22

**Posted:** Mon Oct 19, 2009 2:04 pm

mavropnevma

My intention was to scorn the futility of such problems, once Catalan's (or equivalently, other strong) theorem has been proved. Is there any interest in asking, for example, to exhibit a $100$ -long arithmetic progression made of primes only, after Tao's result? (I think the longest known before that was $23$ -long or whereabouts).

As for the triviality of the problem as stated, I will quote from Sierpinski's 250 problems of number theory (albeit a venerable piece of work)

**Posted:** Mon Oct 19, 2009 2:38 pm

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rex88 · P versus NP Offline Joined: 12 Sep 2009 Posts: 22

**Posted:** Wed Oct 21, 2009 2:55 am