Primes with small primitive roots
Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 1, pp. 71-89

Let $\delta (p)$ tend to zero arbitrarily slowly as $p\rightarrow \infty $. We exhibit an explicit set $\mathcal{S}$ of primes $p$, defined in terms of simple functions of the prime factors of $p-1$, for which the least primitive root of $p$ is $\le p^{1/4-\delta (p)}$ for all $p\in \mathcal{S}$, where $\#\lbrace p\le x: p\in \mathcal{S}\rbrace \sim \pi (x)$ as $x\rightarrow \infty $.

Supposons que la fonction $\delta (p)$ tend vers zéro arbitrairement lentement quand $p\rightarrow \infty $. Nous présentons un ensemble explicite $\mathcal{S}$ de nombres premiers $p$, défini en termes de fonctions simples des facteurs premiers de $p-1$, pour lequel la racine primitive la plus petite de $p$ est $\le p^{1/4-\delta (p)}$ pour tout $p\in \mathcal{S}$, où $\#\lbrace p\le x : p\in \mathcal{S}\rbrace \sim \pi (x)$ quand $x\rightarrow \infty $.

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DOI : 10.5802/jtnb.1355
Classification : 11A07, 11A15, 11A41
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Kevin Ford; Mikhail R. Gabdullin; Andrew Granville. Primes with small primitive roots. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 1, pp. 71-89. doi: 10.5802/jtnb.1355
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[1] Nesmith C. Ankeny The least quadratic non-residue, Ann. Math., Volume 55 (1952), pp. 65-72 | Zbl | DOI

[2] D. A. Burgess The distribution of quadratic residues and non-residues, Mathematika, Volume 4 (1957), pp. 106-112 | Zbl | DOI | MR

[3] D. A. Burgess; Peter D. T. A. Elliott The average of the least primitive root, Mathematika, Volume 15 (1968) no. 1, pp. 39-50 | Zbl | DOI | MR

[4] Richard Crandell; Carl Pomerance Prime Numbers: a Computational Perspective, Springer, 2005, xvi+597 pages | Zbl | MR

[5] Paul Erdős On the normal number of prime factors of p-1 and some related problems concerning Euler’s ϕ-function, Q. J. Math., Volume 6 (1935), pp. 205-213 | Zbl | DOI

[6] Paul Erdős; Harold N. Shapiro On the least primitive root of a prime, Mathematics, Volume 7 (1957), pp. 861-865 | Zbl

[7] Kevin Ford Poisson Approximation of Prime Divisors of Shifted Primes, Int. Math. Res. Not., Volume 2025 (2025) no. 7, rnaf079, 16 pages | MR | Zbl

[8] Kevin Ford; Moubariz Garaev; Sergei Konyagin On the smallest simultaneous power nonresidue modulo a prime, Forum Math., Volume 29 (2017) no. 2, pp. 347-355 | DOI | Zbl | MR

[9] Gabor Halasz Remarks to my paper: “On the distribution of additive and the mean values of multiplicative arithmetic functions”, Acta Math. Acad. Sci. Hung., Volume 23 (1972), pp. 425-432 | Zbl | DOI | MR

[10] Richard R. Hall; Gérald Tenenbaum Divisors, Cambridge Tracts in Mathematics, 90, Cambridge University Press, 1988, xvi+167 pages | Zbl | DOI | MR

[11] Adolf Hildebrand; Gérald Tenenbaum Integers without large prime factors, J. Théor. Nombres Bordeaux, Volume 5 (1993) no. 2, pp. 411-484 | Zbl | DOI | Numdam | MR

[12] Loo-Keng Hua On the least primitive root of a prime, Bull. Am. Math. Soc., Volume 48 (1942), pp. 726-730 | Zbl | MR

[13] Aleksandar Ivić The Riemann zeta-function. Theory and Applications, Theory and applications, John Wiley & Sons, 1985, xvi+517 pages | Zbl | MR

[14] Henryk Iwaniec On the problem of Jacobsthal, Demonstr. Math., Volume 11 (1978), pp. 225-231 | Zbl | DOI | MR

[15] Yurii V. Linnik A remark on the least quadratic non-residue, C. R. (Dokl.) Acad. Sci. URSS, n. Ser., Volume 36 (1942), pp. 119-120 | Zbl | MR

[16] Greg Martin The least prime primitive root and the shifted sieve, Acta Arith., Volume 80 (1997) no. 3, pp. 277-288 | Zbl | DOI | MR

[17] Karl K. Norton On the number of restricted prime factors of an integer. I, Ill. J. Math., Volume 20 (1976) no. 4, pp. 681-705 | Zbl | MR

[18] Andrea Sartori Least primitive root and simultaneous power non-residues, J. Number Theory, Volume 204 (2019), pp. 246-263 | Zbl | DOI | MR

[19] Victor Shoup Searching for primitive roots in finite fields, Math. Comput., Volume 58 (1992), pp. 369-380 extended abstract in the Proceedings of the 22nd ACM Symposium on Theory of Computation (STOC), pp. 546–554, 1990 | Zbl | DOI | MR

[20] Gérald Tenenbaum Introduction to analytic and probabilistic number theory, Graduate Studies in Mathematics, 163, American Mathematical Society, 2015, xxiv+629 pages (translated from the 2008 French edition by Patrick D. F. Ion.) | Zbl | DOI | MR

[21] I. M. Vinogradov On the distribution of quadratic residues and nonresidues, J. Phys.-Math. Soc. Perm (1919)

[22] I. M. Vinogradov On the bound of the least non-residue of nth powers, Trans. Am. Math. Soc., Volume 29 (1927), pp. 218-226 | Zbl | MR

[23] I. M. Vinogradov On the least primitive root, C. R. Acad. Sci. URSS, Volume 1930 (1930), pp. 7-11 | Zbl

[24] Yuan Wang A note on the least primitive root of a prime, Sci. Record, New Ser., Volume 3 (1959), pp. 174-179 | Zbl | MR

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