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A014127 Mirimanoff primes: primes p such that p^2 divides 3^(p-1) - 1. 36
11, 1006003 (list; graph; refs; listen; history; text; internal format)



Dorais and Klyve proved that there are no further terms up to 9.7*10^14.

These primes are so named after the celebrated result of Mirimanoff in 1910 (see below) that for a failure of the first case of Fermat's Last Theorem, the exponent p must satisfy the criterion stated in the definition. Lerch (see below) showed that these primes also divide the numerator of the harmonic number H(floor(p/3)). This is analogous to the fact that Wieferich primes (A001220) divide the numerator of the harmonic number H((p-1)/2). - John Blythe Dobson, Mar 02 2014, Apr 09 2015


Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, 1979, pp. 23, 152-153.

Alf van der Poorten, Notes on Fermat's Last Theorem, Wiley, 1996, p. 21.


Table of n, a(n) for n=1..2.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.

C. K. Caldwell, Fermat Quotient, The Prime Glossary.

F. G. Dorais and D. Klyve, A Wieferich prime search up to  p < 6.7*10^15, J. Integer Seq. 14 (2011), Art. 11.9.2, 1-14.

W. Keller, J. Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.

M. Lerch, Zur Theorie des Fermatschen Quotienten..., Mathematische Annalen 60 (1905), 471-490.

D. Mirimanoff, Sur le dernier théorème de Fermat, C. R. Acad. Sci. Paris, 150 (1910), 204-206. Revised as Sur le dernier théorème de Fermat, Journal für die reine und angewandte Mathematik 139 (1911), 309-324.

Planet Math, Wieferich Primes


Select[Prime[Range[1000000]], PowerMod[3, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)



N=10^9; default(primelimit, N);

forprime(n=2, N, if(Mod(3, n^2)^(n-1)==1, print1(n, ", ")));

\\ Joerg Arndt, May 01 2013


from sympy import prime

from gmpy2 import powmod

A014127_list = [p for p in (prime(n) for n in range(1, 10**7)) if powmod(3, p-1, p*p) == 1] # Chai Wah Wu, Dec 03 2014


Cf. A001220, A039951, A096082.

Sequence in context: A253632 A112854 A211238 * A049192 A156670 A116061

Adjacent sequences:  A014124 A014125 A014126 * A014128 A014129 A014130




N. J. A. Sloane


Edited by Max Alekseyev, Oct 20 2010

Updated by Max Alekseyev, Jan 29 2012



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