
OFFSET

1,1


COMMENTS

Sometimes called Mirimanoff primes.  Matthijs Coster, Jun 30 2008
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((p1)/2).  John Blythe Dobson, Mar 02 2014, Apr 09 2015


REFERENCES

Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, 1979, pp. 23, 152153.
Alf van der Poorten, Notes on Fermat's Last Theorem, Wiley, 1996, p. 21.


LINKS

Table of n, a(n) for n=1..2.
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, 114.
W. Keller, J. Richstein, Solutions of the congruence a^(p1) == 1 (mod p^r), Math. Comp. 74 (2005), 927936.
M. Lerch, Zur Theorie des Fermatschen Quotienten..., Mathematische Annalen 60 (1905), 471490.
D. Mirimanoff, Sur le dernier théorème de Fermat, C. R. Acad. Sci. Paris, 150 (1910), 204206. Revised as Sur le dernier théorème de Fermat, Journal für die reine und angewandte Mathematik 139 (1911), 309324.
Planet Math, Wieferich Primes


PROG

(PARI)
N=10^9; default(primelimit, N);
forprime(n=2, N, if(Mod(3, n^2)^(n1)==1, print1(n, ", ")));
\\ Joerg Arndt, May 01 2013
(Python)
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, p1, p*p) == 1] # Chai Wah Wu, Dec 03 2014


CROSSREFS

Cf. A001220, A039951, A096082.
Sequence in context: A253632 A112854 A211238 * A049192 A156670 A116061
Adjacent sequences: A014124 A014125 A014126 * A014128 A014129 A014130


KEYWORD

nonn,hard,bref,more


AUTHOR

N. J. A. Sloane


EXTENSIONS

Edited by Max Alekseyev, Oct 20 2010
Updated by Max Alekseyev, Jan 29 2012


STATUS

approved

