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OFFSET
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1,1
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COMMENTS
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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
The prime 1006003 was apparently discovered by K. E. Kloss (cf. Kloss, 1965) according to various sources. - Felix Fröhlich, Dec 08 2020
If there is no term other than 11 and 1006003, then the only solution (a, w, x, y, z) to the diophantine equation a^w + a^x = 3^y + 3^z is (5, 1, 1, 2, 3) (cf. Scott, Styer, 2006, Lemma 12). - Felix Fröhlich, Dec 10 2020
Named after the Russian mathematician Dmitry Semionovitch Mirimanoff (1861-1945). - Amiram Eldar, Jun 10 2021
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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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K. E. Kloss, Some Number-Theoretic Calculations, Journal of Research of the National Bureau of Standards - B. Mathematics and Mathematical Physics, Vol. 69B, No. 4 (Oct.-Dec. 1965), pp. 335-336.
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MATHEMATICA
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Select[Prime[Range[1000000]], PowerMod[3, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)
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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, ", ")));
(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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Sequences "primes p such that p^2 divides X^(p-1)-1": A001220 (X=2), A123692 (X=5), A212583 (X=6), A123693 (X=7), A045616 (X=10), A111027 (X=12), A128667 (X=13), A234810 (X=14), A242741 (X=15), A128668 (X=17), A244260 (X=18), A090968 (X=19), A242982 (X=20), A298951 (X=22), A128669 (X=23), A306255 (X=26), A306256 (X=30).
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KEYWORD
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nonn,hard,bref,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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