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It is known that the only nonzero pairwise coprime integer solutions to the above Diophantine equation are for $r=3$, for which $(x, y, z) = (3,1,1)$ and $(-3,1, 1)$. (Cohen, Number Theory Volume 2: Analytic and Modern Tools pp 507-509, Theorem 15.3.4).

When one only assumes that $\gcd(x, y, z)=1$, we also have the solution $(x, y, z) = (3\cdot2^{\frac{r}{2}},2,1)$, for $r=p-3$. But are there any other nonzero integral solutions with none of $(\mid x \mid, \mid y \mid, \mid z \mid)$ being equal to $1$ ?

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    $\begingroup$ You mention this about the tags each time you ask a question. But I see no reason why you shouldn't be able to add the tags yourself. $\endgroup$ – Lucia Dec 9 '15 at 22:29
  • $\begingroup$ @Lucia, Well, each time i try to use the relevant tags, i get a notice that such and such a tag requires a certain minimum reputation (which i currently don't have since i'm still new here). I would indeed be grateful if you may share with me on how i can go about it. $\endgroup$ – user83236 Dec 9 '15 at 22:36
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    $\begingroup$ I just checked it out from a different browser. I didn't encounter any problems in adding any tags to a fake question (while not logged into Mathoverflow). You could ask this in Meta, if indeed you're having some difficulties with the site. $\endgroup$ – Lucia Dec 9 '15 at 22:39
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    $\begingroup$ @Tatenda: You can use any existing tag, also nt.number-theory. -- You only cannot create new tags, as this requires 300 points, cf. mathoverflow.net/help/privileges/create-tags. $\endgroup$ – Stefan Kohl Dec 9 '15 at 22:39
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Well, i have just found a certain paper on this subject by W. Ivorra of 2003 on the Acta Arithmetica website. Unfortunately, it is written in French, which i'm not so fluent in and Google Translator is not offering much help. Therein i found a theorem that says:

"Nous definissons de meme le fait pour $(a, b, c)$ de etre une solution propre non trivile de l'equation

Theoreme (a): Let $S_{0}(r,p)$ denote the set of all nonzero integral solutions to the Diophantine equation concerned. (this was the only part that i could translate into English). Si $r$ distinct de $1, 3, p-3$ et $p-1$, l'énsemble $S_{0} (r,p)$ est vide."

From my very limited understanding of French, i think its saying that $S_0(r, p)$ is empty if $r$ is not equal to $1,3, p-1$ or $p-3$.
Any corrections from more French-fluent OP's are most welcome.

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  • $\begingroup$ You're translation is correct. $\endgroup$ – user78249 May 13 '16 at 21:36
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There is a nice article by Samir Siksek: Siksek-2003.

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