# How do you bound the exponent of x^2+1=y^p?

How do you bound the exponent of $x^2+1=y^p$ for $p$ a prime exponent using linear forms in logs?

So far I have $(x-i)(x+i)=y^p$ which are coprime and hence $x+i=(a+ib)^p$.

Now how do I get a linear form in logs so that I can find an upper bound on $p$?

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– Will Jagy Jun 15 '12 at 19:43
not what link is supposed to mean, I'm not asking for a full proof of cataln's conjecture but consider this particular e.g. – Kale Jun 15 '12 at 20:34
perhaps you can give some references that use the techniques you want to apply. – Will Jagy Jun 15 '12 at 23:53
You can look up the work of Tijdeman for this kind of argument. But why would you not want to use a known theorem? – Felipe Voloch Jun 16 '12 at 0:05

If $x+i=(a+ib)^p$, then $x-i=(a-ib)^p$, $2i=(a+ib)^p-(a-ib)^p=2ib$(smth), $b=\pm 1$, $1=\pm 1\mp \binom{p}2a^2\pm \binom{p}4 a^4+\dots$. Considering this modulo $a^2$ we see that signs start from plus, that is, $\binom{p}2a^2\pm \binom{p}4 a^4+\dots=0$. Next, $\binom{p}{2k}=\binom{p}2\binom{p-2}{2k-2}/\binom{2k}2$ and dividing by $\binom{p}2a^2$ we get $$1=\sum_{k\geqslant 2} \pm \binom{p-2}{2k-2}\frac{a^{2k-2}}{k(2k-1)}.$$ If $a$ has some prime divisor $q$, then RHS has positive $q$-adic valuation, since $k(2k-1)$ can not be divisible by $q^{2k-2}$. If $a=\pm 1$, then we know what is $(a+i)^p$ and it is not $x+i$.