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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, 2013 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, 2013 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, 2013 at 8:43
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    $\begingroup$ @AndréNicolas,yes,$-1,0,1$ are 3 consecutive powers. $\endgroup$
    – rah4927
    Dec 17, 2013 at 9:01

2 Answers 2

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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, 2013 at 8:25
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    $\begingroup$ Funny how such simple statements have extremely difficult proofs. $\endgroup$
    – rah4927
    Dec 17, 2013 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, 2013 at 8:53
  • $\begingroup$ @GottfriedHelms "Theorem: <complicated hieroglyphics> Proof: Left to reader." $\endgroup$
    – Akiva Weinberger
    Oct 29, 2015 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, 2015 at 9:41
  • $\begingroup$ @user21820 Mihăilescu's proof appeared in Crelle's Journal in 2004. $\endgroup$
    – Konstantinos Gaitanas
    Oct 29, 2015 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, 2015 at 11:49

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