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Statement

Notation (p. 11): $r(p)$ is the least primitive root of the prime $p$. The passage follows conjecture (4) and states, in this order, without proof:

1. A proof of

$$\sum_{p<x}r(p)=\bigl(1+o(1)\bigr)\,c\,\frac{x}{\log x}$$

seems very hard. 2. It has not even been proved that $r(p)$ does not tend to infinity with $p$. 3. Artin conjectured that $2$ is a primitive root of infinitely many primes, and a proof seems very hard. 4. As far as the author knows, it has not even been proved that every prime $p$ has a prime $q<p$ that is a primitive root of $p$. In the paper's words: "Tudtommal még az sincs bebizonyítva, hogy minden $p$ prímszámhoz van oly $q<p$ prímszám, mely $p$-nek primitív gyöke." 5. After reviewing upper bounds for $r(p)$ (Vinogradov's (5), $r(p)<p^{1/2+\varepsilon}$ for $p>p_0(\varepsilon)$; the improvement of Hua, H. Shapiro and the author to $r(p)<c_3p^{1/2}v(p-1)^{c_4}$, with $v(p-1)$ the number of distinct prime factors of $p-1$; and a sharper power bound of Burgess and Wang), the paper says it may well be that $r(p)<c\log p$.

Item 4 quantifies over every prime $p$, as printed; for $p=2$ there is no prime below $p$.

Source. P. Erdős, Számelméleti megjegyzések, I. (Remarks on number theory, I.; in Hungarian), Mat. Lapok 12 (1961), 10--17; MR 26 #2410, Zbl 0154.294. All five remarks on printed p. 11, read on the page image of the edition identified on the source card. The exponent in the displayed Burgess--Wang bound is not legible on the scan and is not recorded here.

Read depth. Claims checked: the passage was read clause by clause on the page image. It contains no proofs.

Bears on

• Problem 985: item 4 is the problem's question in the site's wording (every prime $p$), which the problem page cites at p. 11; the problem page's corrected Statement asks it for every prime $p>2$. The paper records it as not known and proves nothing about it.