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Let $p$ and $q$ be prime numbers such that $p^2+p+1=3q^a$: is it true that $a=1$?

This specific equation appears when computing order components of finite groups.

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Not a complete answer but an indication of what is known:

The solutions to your diophantine equation known due to Nagell. I have in front of me Paulo Ribenboim's "My Numbers, My Friends" from which I quote:

Theorem. If $m > 2$, the only non-zero solutions of $X^2 + X + 1 = 3 Y^m$ are $x = 1$ and $x = -2$. If $m = 2$, there are also the solutions $$x = \pm \frac{\sqrt{3}}{4}\left((2+\sqrt{3})^{2n+1} - (2-\sqrt{3})^{2n+1}\right) - \frac{1}{2}$$ for $n =0, 1, \dots$.

The $m = 2$ case is quite clear: multiply through by $4$ and note that the given equation is equivalent to $$(2X+1)^2 - 3(2Y)^2 = -3$$ Put $X' = 2Y$ and $Y' = (2X+1)/3$ so that you get $$X'^2 - 3Y'^2 = 1$$ and $2 - \sqrt{3}$ is the fundamental unit in the real quadratic field $\mathbf{Q}(\sqrt{3})$.

The reference from Paulo's book is:

T. Nagell. Des équations indéterminées $x^2 + x + 1 = y^n$ et $x^2 + x + 1 = 3y^n$. Norsk Mat. Forenings Skrifter, Ser. I, 1921, No. 2, 14 pages. (= 1921a at the end of Chapter 7.)

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  • $\begingroup$ In the Pell like case, how do you decide whether $x$ is prime or not, and if so, whether $y$ is also prime? $\endgroup$
    – Geoff Robinson
    Commented May 19, 2015 at 15:31
  • $\begingroup$ I still don't see how to do this with "prime" condition. $\endgroup$
    – knsam
    Commented May 19, 2015 at 16:09
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    $\begingroup$ This sequence is similar to Fibonacci, and we don't know whether there are infinitely many Fibonacci primes. And $n=8$ is another prime solution $x=2288805793$, $y=1321442641$, so any proof would need that as a special case. I don't know a strategy for solving this. $\endgroup$
    – Linus Hamilton
    Commented May 19, 2015 at 16:10
  • $\begingroup$ @LinusHamilton: Your remark is very appropriate. Let me add that the sequence $2^n-1$ is very similar, e.g. it satisfies a similar recursion, and it contains infinitely many primes if and only there are infinitely many even perfect numbers. This is one of the oldest unsolved problems in mathematics. $\endgroup$
    – GH from MO
    Commented May 19, 2015 at 16:54
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    $\begingroup$ The probability that $x$ is prime is $O(1/\log x)=O(1/n)$ and since $y\sim x$, the probability that $y$ is prime is the same. So the probability that $x$ and $y$ are both primes is $O(1/n^2)$. (Of course, I'm making some randomness and independence assumptions that can't be justified, but just to get an idea...) Summing, this suggests that there are only finitely many such solutions. $\endgroup$
    – Joe Silverman
    Commented May 19, 2015 at 16:55
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$$313^2 + 313 + 1 = 3\times181^2$$

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    $\begingroup$ "I said nontrivial." (mathoverflow.net/q/27305/2383) $\endgroup$
    – LSpice
    Commented May 19, 2015 at 13:32
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    $\begingroup$ True enough. :) This is the only solution with $p<10^8$. The case $a=2$ should boil down to a Pell-like equation. But I don't want to dig too deep into it without hearing OP's response, as I don't know how (313, 181) and their possible friends apply to finite groups. $\endgroup$
    – Linus Hamilton
    Commented May 19, 2015 at 13:35
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Another counterexample for $a=2$: $p=2288805793$, $q=1321442641$.

There are only 2 counterexamples in primes below $10^{5000}$.

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  • $\begingroup$ $10^{5\times 10^3}$? $\endgroup$
    – Turbo
    Commented May 20, 2015 at 3:08
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    $\begingroup$ @Turbo: Yes. What is your concern? $\endgroup$
    – Max Alekseyev
    Commented May 20, 2015 at 3:11
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    $\begingroup$ @Turbo: Solutions to the Pellian equation (explained above by knsam) grow exponentially fast, so it is not a big deal to test them all below a bound like $10^{5000}$. In fact, I tested smallest $10^4$ solutions and achieved the bound $10^{5722}$. $\endgroup$
    – Max Alekseyev
    Commented May 20, 2015 at 3:21
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    $\begingroup$ This counterexample was in Linus Hamilton's comment on knsam's (not-a-complete-)answer. $\endgroup$
    – Gerry Myerson
    Commented May 20, 2015 at 3:57
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    $\begingroup$ @GerryMyerson: Oh, I did not notice. Anyway, no search bound was specified explicitly so far, but here it comes. $\endgroup$
    – Max Alekseyev
    Commented May 20, 2015 at 4:05

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