# Common primitive roots modulo several primes

Motivated by the Chinese Remainder Theorem, I'm interested in common primitive roots modulo several primes.

Let's first look at common primtive roots modulo two distinct primes. For any distinct primes $p$ and $q$, does there exist a positive integer $g\leqslant \sqrt{4pq+1}$ which is a primitive root modulo $p$ and also a primitive root modulo $q$? In August 2017, I conjectured that the answer is yes. Moreover, I think that we may even replace $\sqrt{4pq+1}$ by $\sqrt{pq}$ if $\{p,q\}$ is not among the following 15 pairs of primes \begin{gather*}\{2,3\},\ \{2,11\},\ \{2,13\},\ \{2,59\},\ \{2,131\}, \ \{2,181\}, \ \{3,7\},\ \{3,31\}, \\\{3,79\},\ \{3,191\},\ \{3,199\},\ \{5,271\}, \ \{7,11\},\ \{7,13\}, \ \{7,71\}. \end{gather*}

In general, for each integer $n>1$, I guess that there is a positve constant $c_n$ such that for any $n$ distinct primes $p_1,\ldots,p_n$ there is a positive integer $g \leqslant c_n\root{n}\of{p_1\cdots p_n}$ which is a primitive root modulo $p_k$ for all $k = 1,\ldots,n$.

See http://oeis.org/A291690 for related data and comments.

Any ideas towards my question on common primitive roots modulo several primes?

• Of possible interest: MR1369278 (96j:11006) Finizio, Norman J.; Lewis, James T.(1-RI); Distribution of common primitive roots. Proceedings of the Twenty-sixth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1995), Congr. Numer. 108 (1995), 85–95. – Gerry Myerson May 21 '18 at 3:28
• Gerry, thank you for pointing out that paper. However, I cannnot find the paper and the MR review does not mention any concrete results or conjectures in the paper. – Zhi-Wei Sun May 22 '18 at 5:18
• Finizio is Professor Emeritus at University of Rhode Island. – Gerry Myerson May 22 '18 at 6:54