# On primitive roots of the form $5^k+10^m$ with $k$ and $m$ nonnegative integers

Let $p$ be any prime. It is well known that the set $$G_p=\{0<g<p:\ g\ \text{is a primitive root modulo}\ p\}$$ has cardinality $\varphi(p-1)$, where $\varphi$ is Euler's totient function. It is interesting to investigate the structure of the set $G_p$. In 2014 I conjectured that $G_p$ always contains a number of the form $x^2+1$ with $x$ an integer (cf. http://oeis.org/A239957). This is still open. Here I ask a further question.

QUESTION: Is it true that for each prime $p>7$ the set $G_p$ always contains a number of the form $5^k+10^m$ with $k,m\in\mathbb N=\{0,1,2,\ldots\}$?

I conjecture that the answer is affirmative, i.e., for any prime $p>7$ there are $k,m\in\mathbb N$ such that $5^k+10^m$ is not only smaller than $p$ but also a primitive root modulo $p$. I have verified this for all primes $p$ with $7<p<10^9$. For the number of ordered pairs $(k,m)\in\mathbb N^2$ with $5^k+10^m\in G_p$, one may visit http://oeis.org/A305048. For example, for the prime $p=6276271$, the integer $$5^5+10^1=3135$$ is the unique element of $G_p$ in the form $5^k+10^m$ with $k,m\in\mathbb N$.

My above question looks somewhat curious and quite challenging. Any comments are welcome!

• Is there any particular reason why you chose the numbers 5 and 10? It seems like one could easily generate a large number of very similar but not very inspiring conjectures, just by choosing combinations of series with reasonable asymptotics. – S. Carnahan May 26 '18 at 14:06
• @S. Camahan I did test various combinations of two powers. $5^k+10^m$ is the most strikiing one since it grows fast and also 5 divides 10. I don't like conjectures with some exceptions greater than 10. – Zhi-Wei Sun May 26 '18 at 22:11