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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?

GH from MO
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darya
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2 Answers2

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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 $p=18$, $q=7$, $a=3$.

GH from MO
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    Just wanted to point out that Ljunggren's first part of the paper you refer to also solves the case $m = 2$ completely too. There was a recent question about this here on MO and I posted there a scan of the paper you cite. – knsam May 18 '15 at 23:55
  • @knsam: Thank you, especially for the scan of Ljunggren's paper. In fact I emphasized the case $m=2$ in my earlier related post (http://mathoverflow.net/questions/177952/when-is-qk-1-q-1-a-perfect-square/177958#177958), 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. – GH from MO May 19 '15 at 00:03
  • One of the perks of being here at Queen's is that I have access to Prof. Paulo Ribenboim's remarkable collection of mathematical papers and books. I will see if I can access a copy of the book and post here tomorrow. :-) – knsam May 19 '15 at 00:29
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    Yes, actually Ho notes at the end of his paper that his Theorem A is covered by Ljunggren's work. – Geoff Robinson May 19 '15 at 00:38
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    @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. – GH from MO May 19 '15 at 00:55
  • Thanks everyone for your help. Also I think if p^2+p+1=3q^a, where q and p are primes, then a=1. Is it true? – darya May 19 '15 at 10:09
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    @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. – Peter Mueller May 19 '15 at 11:53
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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).

Ryan Dougherty
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Geoff Robinson
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