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OFFSET
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1,1
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COMMENTS
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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((p-1)/2). - John Blythe Dobson, Mar 02 2014, Apr 09 2015
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REFERENCES
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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.
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LINKS
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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, 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
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PROG
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(PARI)
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
(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, p-1, p*p) == 1] # Chai Wah Wu, Dec 03 2014
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CROSSREFS
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Cf. A001220, A039951, A096082.
Sequence in context: A253632 A112854 A211238 * A049192 A156670 A116061
Adjacent sequences: A014124 A014125 A014126 * A014128 A014129 A014130
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KEYWORD
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nonn,hard,bref,more
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Edited by Max Alekseyev, Oct 20 2010
Updated by Max Alekseyev, Jan 29 2012
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STATUS
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approved
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