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On Grosswald’s conjecture on primitive roots

Volume 172 / 2016

Stephen D. Cohen, Tomás Oliveira e Silva, Tim Trudgian Acta Arithmetica 172 (2016), 263-270 MSC: Primary 11L40; Secondary 11A07. DOI: 10.4064/aa8109-12-2015 Published online: 16 December 2015


Grosswald’s conjecture is that $g(p)$, the least primitive root modulo $p$, satisfies $g(p) \leq \sqrt{p} - 2$ for all $p>409$. We make progress towards this conjecture by proving that $g(p) \leq \sqrt{p} -2$ for all $409 < p < 2.5\times 10^{15}$ and for all $p > 3.38\times 10^{71}$.


  • Stephen D. CohenSchool of Mathematics and Statistics
    University of Glasgow
    Glasgow G12 8QW, Scotland
  • Tomás Oliveira e SilvaDepartamento de Electrónica,
    Telecomunicações e Informática
    Universidade de Aveiro
    3810-193 Aveiro, Portugal
  • Tim TrudgianMathematical Sciences Institute
    The Australian National University
    Canberra, ACT 2601, Australia

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