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I have been wondering whether consecutive integers can ever be perfect powers.And even if they can, how many consecutive integers at most can be perfect powers?My intuition tells me that consecutive integers can never be perfect powers,but I don't want to let that cloud my judgement.I haven't done any work,mainly because I don't know where to start.A hint that would help me start my proof will be appreciated.

EDIT: 8 and 9 clearly are perfect powers.I didn't know that it is called Catalan's conjecture.

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    $\begingroup$ What about 8, 9? $\endgroup$ – Harald Hanche-Olsen Dec 17 '13 at 8:22
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    $\begingroup$ The fact that it was only proved in 2002 should give you a hint that the proof is far from easy. $\endgroup$ – Harald Hanche-Olsen Dec 17 '13 at 8:26
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    $\begingroup$ If we are talking about integers, there is also $-1,0$ and $0,1$. $\endgroup$ – André Nicolas Dec 17 '13 at 8:43
  • $\begingroup$ @AndréNicolas,yes,$-1,0,1$ are 3 consecutive powers. $\endgroup$ – rah4927 Dec 17 '13 at 9:01
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The only case of consecutive perfect powers is 8 and 9. This is Catalan's conjecture, which was proven to be true.

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  • $\begingroup$ Ah. I should have remembered that! I took the liberty of adding a link to Wikipedia. $\endgroup$ – Harald Hanche-Olsen Dec 17 '13 at 8:25
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    $\begingroup$ Funny how such simple statements have extremely difficult proofs. $\endgroup$ – rah4927 Dec 17 '13 at 8:26
  • $\begingroup$ @rah: consider the converse: a simple proof, but the problem is complicated to formulate... ;-) (just kidding) $\endgroup$ – Gottfried Helms Dec 17 '13 at 8:53
  • $\begingroup$ @GottfriedHelms "Theorem: <complicated hieroglyphics> Proof: Left to reader." $\endgroup$ – Akiva Weinberger Oct 29 '15 at 13:46
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The fact that there exist finitely many was first proved by Tidjeman and a complete proof that $8=2^3$ and $9=3^2$ is the only pair of consecutive powers was proved by Preda Mihăilescu in 2002 and published in 2004.
The proof is not an easy one to show here.

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  • $\begingroup$ Why does your answer say 2004 but Wikipedia says 2002? $\endgroup$ – user21820 Oct 29 '15 at 9:41
  • $\begingroup$ @user21820 Mihăilescu's proof appeared in Crelle's Journal in 2004. $\endgroup$ – Konstantinos Gaitanas Oct 29 '15 at 10:45
  • $\begingroup$ Yes that's what Wikipedia says too, but I thought we usually state the time it was known to be proven rather than printed. $\endgroup$ – user21820 Oct 29 '15 at 11:49

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