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can you help me please for solving this diophantine equation : $x^2=y^p+1$ and if you can give me a general method to studying such equation : $x^2=y^p+t$

Thanks

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    $\begingroup$ This is not appropriate for this forum. For t=1, you find that x^2 is at most 10, by looking at prime factors of y. In general, you study how squares lie between powers and find stuff studied by Pillai. Mike Bennet (if I remember correctly) can tell you more. Consider asking this on math.stackexchange. Gerhard "Try Elementary Number Theoretic Methods" Paseman, 2015.11.13 $\endgroup$
    – Gerhard Paseman
    Commented Nov 13, 2015 at 21:03
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    $\begingroup$ Can those voting to close think about the problem for a few minutes? If they do see a quick solution, I would certainly appreciate knowing about it. (I can see how a proof would go, as indicated in Zudilin's answer below, but it certainly is not immediate.) $\endgroup$
    – Lucia
    Commented Nov 14, 2015 at 1:14
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    $\begingroup$ Dear OP: if you could provide more context for how this problem arose in your research, it might help alleviate the standard concern of a question not being of "research level". (We do encourage this practice wherever possible.) $\endgroup$
    – Todd Trimble
    Commented Nov 14, 2015 at 1:39

2 Answers 2

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This ($t=1$) is a particular case of the famous Catalan equation. The only solution known is $3^2=2^3+1$. The case $p=3$ was done by Euler and the case $p>3$ was done by Ko Chao in 1964 (the English proof is published in Mordell's book); E. Chein published an elementary (and very nice!) proof of Chao's theorem in 1976 (PAMS 56, pp. 83-84).

The general $t\in\mathbb Z\setminus\{0\}$ (fixed) case is a special case of Pillai's conjecture: it is expected that only finitely many solutions in integers show up.

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  • $\begingroup$ I fixed the date of Ko Chao's publication (Sci. Sinica (Notes) 14 (1964), 457-460). $\endgroup$
    – GH from MO
    Commented Nov 14, 2015 at 0:04
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    $\begingroup$ I don't understand the down-vote to the question, really: if we ignore the specifics, and "make it a better question than it is", it asks about higher-genus curves, Mordell-Weil, ABC-conjecture, and such, I think, ... $\endgroup$
    – paul garrett
    Commented Nov 14, 2015 at 0:59
  • $\begingroup$ @paulgarrett Perhaps the downvote was reversed; right now none is there. $\endgroup$
    – Todd Trimble
    Commented Nov 14, 2015 at 1:40
  • $\begingroup$ @WadimZudilin Chein's paper cites Nagell's previous result that $p$ divides $y+1$, I can't neither find Nagell's paper online nor prove it myself, would you please explain this? $\endgroup$
    – Fedor Petrov
    Commented Nov 14, 2015 at 11:46
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    $\begingroup$ Fedor, this divisibility result is in the heart of the resolution of Catalan's problem by Preda Mihăilescu, not just of this particular case. It is in Mordell's book but I also have all the related things recorded in Russian (wain.mi.ras.ru/cp) after our 2001-02 Moscow seminar. The notes ("Elementary introduction") includes Euler's proof, Nagell's results and proofs from both Ko Chao and Chein. It is a fascinating story! $\endgroup$
    – Wadim Zudilin
    Commented Nov 14, 2015 at 12:17
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For $t = 1$, your question is about a special case of Catalan's conjecture, which has been proved in 2002 by Preda Mihăilescu. In particular, for $t = 1$ the only solution is $3^2 = 2^3 + 1$.

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