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Let $n$ a prime, and let $\mathbb{Z}_n$ denote the integers modulo $n$. Let $\mathbb{Z}^*_n$ denote the multiplicative group of $\mathbb{Z}_n$

Are there infinitely many $n$ such that $\mathbb{Z}^*_n$ is generated by $\{ -1, 2 \}$?

Artin's conjecture on primitive roots implies something even stronger: that there are infinitely many $n$ such that $\mathbb{Z}^*_n$ is generated by $\{ 2 \}$. Although likely to be true (in particular it is implied by the Generalized Riemann Hypothesis), as far as I know this conjecture remains open. I am wondering if it is possible that with generators $\{-1,2 \}$, this is known unconditionally.

(One could, of course, ask this for any two generators. For reasons that I'll omit here, I am especially interested in the the generating set $\{-1,2 \}$.)

Matthew Kahle
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    It is certainly easier with extra generators; if you allow three prime generators the result is know. See the result by Heath-Brown on the page you linked to. I have doubts that $-1$ will be of much help though, but I might be wrong. It definitely cannot help for primes conngruent $1$ mod $4$ as then the group has order divisible by $4$ so adding the element $-1$ which of order $2$ can never augement the subgroup generated by $2$ to the full group. – quid Jul 18 '16 at 00:19

2 Answers2

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I am afraid this is out of reach. As quid comments, one can do this with three prime generators, but even two prime generators is too hard; and including $-1$ as a generator does not help much.

Community
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user345872
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The subgroup $\langle -1, 2\rangle$ generated by $-1$ and $2$ is either generated by $2$ or by $-2$, depending on the parity of the order of $2$. The generator of $\langle -1, 2\rangle$ is not certain, making the study complicated.

However, if we consider the order of $\langle -1, 2\rangle$, it is related to the order of $4$, a specific element that can always be used. Specifically, $|\langle -1, 2\rangle|$ is equal to $2 \cdot |\langle 4 \rangle|$ in $\mathbb{Z}_n^*$ for every odd prime $n$ (thanks to Professor Pieter Moree). The question then becomes whether there are infinitely many primes $n$ such that $|\langle 4 \rangle| = (n-1)/2$. In some sense, the original question still concerns a subgroup generated by a specific element. From this perspective, $-1$ does not seem to be very helpful.

By the way, the relation $|\langle -1, 2\rangle| = 2 \cdot |\langle 4 \rangle|$ is useful for computing the natural density of primes $n$ such that the index of $\langle -1, 2\rangle$ in $\mathbb{Z}_n^*$ is a fixed number, assuming the generalized Riemann hypothesis holds. One may find more details in my note 2310.04527 on arXiv.

Wei-Liang Sun
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