A new prime 𝑝 for which the least primitive root (𝑚𝑜𝑑𝑝) and the least primitive root (𝑚𝑜𝑑𝑝²) are not equal

Paszkiewicz, A.

doi:10.1090/S0025-5718-08-02090-5

A new prime $p$ for which the least primitive root $(\textrm {mod} p)$ and the least primitive root $(\textrm {mod} p^2)$ are not equal
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by A. Paszkiewicz;

Math. Comp. 78 (2009), 1193-1195

DOI: https://doi.org/10.1090/S0025-5718-08-02090-5

Published electronically: October 31, 2008

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Abstract:

With the aid of a computer network we have performed a search for primes $p<10^{12}$ and revealed a new prime $p=6692367337$ for which its least primitive root $(\textrm {mod} p)$ and its least primitive root $(\textrm {mod} p^2)$ are not equal.

References

N. Beeger, On a new case of the congruence $2^{p-1}\equiv 1 (\textrm {mod} p^2)$. Messenger of Mathematics, 51 (1922), pp. 149–150.
Richard Crandall, Karl Dilcher, and Carl Pomerance, A search for Wieferich and Wilson primes, Math. Comp. 66 (1997), no. 217, 433–449. MR 1372002, DOI 10.1090/S0025-5718-97-00791-6
Wilfrid Keller and Jörg Richstein, Solutions of the congruence $a^{p-1}\equiv 1\pmod {p^r}$, Math. Comp. 74 (2005), no. 250, 927–936. MR 2114655, DOI 10.1090/S0025-5718-04-01666-7
Joshua Knauer and Jörg Richstein, The continuing search for Wieferich primes, Math. Comp. 74 (2005), no. 251, 1559–1563. MR 2137018, DOI 10.1090/S0025-5718-05-01723-0
E. L. Litver and G. E. Judina, Primitive roots for the first million primes and their powers, Mathematical analysis and its applications, Vol. III (Russian), Izdat. Rostov. Univ., Rostov-on-Don, 1971, pp. 106–109 (Russian). MR 340159
W. Meissner, Über die Teilbarkeit von $2^p-2$ durch das Quadrat der Primzahl $p=1093$, Sitzungsberichte Preuss. Akad. Wiss. (1913), pp. 663–667.
Peter L. Montgomery, New solutions of $a^{p-1}\equiv 1\pmod {p^2}$, Math. Comp. 61 (1993), no. 203, 361–363. MR 1182246, DOI 10.1090/S0025-5718-1993-1182246-5
A. Paszkiewicz and A. Schinzel, Numerical calculation of the density of prime numbers with a given least primitive root, Math. Comp. 71 (2002), no. 240, 1781–1797. MR 1933055, DOI 10.1090/S0025-5718-01-01382-5