Let $p$ be a prime number, I think when $p^2+p+1=q^a$, where $q$ is a prime number, then $a=1$. But I can't prove it. Is it true?
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2$\begingroup$ Please use TeX on this site. $\endgroup$– GH from MOCommented May 18, 2015 at 23:29
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2 Answers
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This question is answered (affirmatively and somewhat more generally) in the following paper: Chat Yin Ho, Projective planes with a regular collineation group and a question about powers of a prime, J. Algebra 154 (1993), no. 1, 141–151. The proof there uses the ring of Eisenstein integers. (The issue is available online via Open Access).
answered May 18, 2015 at 16:03
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$\begingroup$ Thanks a lot Mr Geoff Robinson, I saw the paper and it was exactly what I wanted. Very nice $\endgroup$– daryaCommented May 18, 2015 at 16:22
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$\begingroup$ Excuse me, is it true for p^2-p+1? $\endgroup$– daryaCommented May 18, 2015 at 16:50
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$\begingroup$ I am not sure. It may be that Ho's proof can be adapted to that case. If it does not, it may point to a counterexample. $\endgroup$ Commented May 18, 2015 at 16:57
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$\begingroup$ @ Geoff Robinson thank you very much for your help and advice. $\endgroup$– daryaCommented May 18, 2015 at 17:28
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10$\begingroup$ @darya $19^2 - 19 + 1 = 7^3$ $\endgroup$ Commented May 18, 2015 at 19:02
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The equation $$ \frac{x^k-1}{x-1}=y^m$$ is known as the Nagell-Ljunggren equation. It is conjectured that for $x\geq 2$, $y\geq 2$, $k\geq 3$, $m\geq 2$, the only solutions are $$ \frac{3^5-1}{3-1}=11^2,\qquad \frac{7^4-1}{7-1}=20^2,\qquad \frac{18^3-1}{18-1}=7^3.$$ For $3\mid k$, the equation was solved by Ljunggren (Norsk. Mat. Tidsskr. 25 (1943), 17-20). For more details see also here.
It follows that $p^2+p+1=q^a$ for any integers $p,q,a\geq 2$ implies $p=18$, $q=7$, $a=3$.
answered May 18, 2015 at 23:44
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$\begingroup$ @knsam: Thank you, especially for the scan of Ljunggren's paper. In fact I emphasized the case $m=2$ in my earlier related post (mathoverflow.net/questions/177952/…), and the linked paper by Bugeaud and Mihailescu mentions other solved cases (by Nagell-Ljunggren) as well. BTW I would love to read a complete account of these results in English. Under my other MO response I linked here, it was remarked that Ribenboim's book about Catalan's conjecture discusses some cases in detail (starting with page 110), but I don't have access to this book. $\endgroup$ Commented May 19, 2015 at 0:03
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1$\begingroup$ Yes, actually Ho notes at the end of his paper that his Theorem A is covered by Ljunggren's work. $\endgroup$ Commented May 19, 2015 at 0:38
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1$\begingroup$ @knsam: Having a full copy of the book would be wonderful. Having the relevant pages on Nagell-Ljunggren only (pp.110-) would be great, too! Thanks for your efforts. $\endgroup$ Commented May 19, 2015 at 0:55
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1$\begingroup$ @darya: Instead of asking a new question in a comment, you probably should pose this as another question. Also, you should tell why you are interested in this specific equation. $\endgroup$ Commented May 19, 2015 at 11:53