A014127 Mirimanoff primes: primes p such that p^2 divides 3^(p-1) - 1.

11, 1006003

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• bref
• more

Comments

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

References

• Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, 1979, pp. 23, 152-153.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 233.
• Alf van der Poorten, Notes on Fermat's Last Theorem, Wiley, 1996, p. 21.

Links

• Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
• Chris K. Caldwell, Fermat Quotient, The Prime Glossary.
• John Blythe Dobson, On the special harmonic numbers H_floor(p/9) and H_floor(p/18) modulo p, arXiv:2302.02027 [math.NT], 2023.
• François G. Dorais and Dominic Klyve, A Wieferich prime search up to p < 6.7*10^15, J. Integer Seq., Vol. 14 (2011), Article 11.9.2, 1-14.
• Wilfrid Keller and Jörg Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp., Vol. 74, No. 250 (2005), pp. 927-936.
• 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.
• Mathias Lerch, Zur Theorie des Fermatschen Quotienten (a^(p-1) - 1)/p == q(a), Mathematische Annalen, Vol. 60 (1905), pp. 471-490.
• D. Mirimanoff, Sur le dernier théorème de Fermat, C. R. Acad. Sci. Paris, Vol. 150 (1910), pp. 204-206. Revised as Sur le dernier théorème de Fermat, Journal für die reine und angewandte Mathematik, Vol. 139 (1911), pp. 309-324.
• Planet Math, Wieferich Primes.
• Reese Scott and Robert Styer, On the generalized Pillai equation +-a^x +-b^y = c, Journal of Number Theory, Vol. 118, No. 2 (2006), pp. 236-265.

Crossrefs

Cf. A039951, A096082.

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).

Programs

Mathematica

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

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

Extensions

Edited by Max Alekseyev, Oct 20 2010

Updated by Max Alekseyev, Jan 29 2012