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(from American Mathematical Monthly, problem 2488. I hope this hasn't been posted before but I'm new and maybe not very good at using the search function effectively..)

Let $p>3$ be a prime. Show that there exists a primitive root $r$ mod $p$ with $0<r<p$ so that $\gcd (r, p-1)=1$.

I've tried a few things, but the main difficulty seems to be that the primitive root $r$ must be reduced mod $p$ and then taken mod $p-1$, and it's hard to do both at the same time. If $d_1<d_2<... < d_{\varphi (p-1)}$ are the totatives of $p-1$, then we want to show that $\{ g^{d_1}, g^{d_2} , ... g^{d_{\varphi (p-1)}} \}$ and $\{ d_1,d_2,..d_{\varphi (p-1)} \}$ have a non-empty intersection but I'm not sure how to proceed at all.

James Li
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1 Answers1

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This problem appeared in "Problems from the book."

I did a little bit of research on this and it seems to be an unsolved problem but don't quote me on this because the source is outdated since this problem was proposed in 1975. (1, 2)

"It has been conjectured that there exists a natural number $k$ where $0<k<p$ which is a primitive root $\pmod p$ and is relatively prime to $p - 1$."

The jstor site tells me that a solution to a special case of the problem appeared in the next volume of American Math Monthly. Unfortunately, I was unable to find the proof on jstor.

Another source is "index to mathematical problems 1975-1979" but it costs quite a lot of money and I was unable to find the solution via the preview on google books.

I'm really surprised it appeared in "Problems from the Book" as "problems for training", seeing how it might not even have a solution.

I hope this helps.

S.C.B.
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James Leng
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