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%I #81 Apr 03 2023 10:36:09
%S 11,1006003
%N Mirimanoff primes: primes p such that p^2 divides 3^(p-1) - 1.
%C Dorais and Klyve proved that there are no further terms up to 9.7*10^14.
%C 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
%C The prime 1006003 was apparently discovered by K. E. Kloss (cf. Kloss, 1965) according to various sources. - _Felix Fröhlich_, Dec 08 2020
%C 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
%C Named after the Russian mathematician Dmitry Semionovitch Mirimanoff (1861-1945). - _Amiram Eldar_, Jun 10 2021
%D Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, 1979, pp. 23, 152-153.
%D Alf van der Poorten, Notes on Fermat's Last Theorem, Wiley, 1996, p. 21.
%H Amir Akbary and Sahar Siavashi, <a href="http://math.colgate.edu/~integers/s3/s3.Abstract.html">The Largest Known Wieferich Numbers</a>, INTEGERS, 18(2018), A3. See Table 1 p. 5.
%H Chris K. Caldwell, <a href="https://t5k.org/glossary/page.php?sort=FermatQuotient">Fermat Quotient</a>, The Prime Glossary.
%H John Blythe Dobson, <a href="https://arxiv.org/abs/2302.02027">On the special harmonic numbers H_floor(p/9) and H_floor(p/18) modulo p</a>, arXiv:2302.02027 [math.NT], 2023.
%H François G. Dorais and Dominic Klyve, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Klyve/klyve3.html">A Wieferich prime search up to p < 6.7*10^15</a>, J. Integer Seq., Vol. 14 (2011), Article 11.9.2, 1-14.
%H Wilfrid Keller and Jörg Richstein, <a href="http://dx.doi.org/10.1090/S0025-5718-04-01666-7">Solutions of the congruence a^(p-1) == 1 (mod p^r)</a>, Math. Comp., Vol. 74, No. 250 (2005), pp. 927-936.
%H K. E. Kloss, <a href="https://doi.org/10.6028/jres.069B.035">Some Number-Theoretic Calculations</a>, Journal of Research of the National Bureau of Standards - B. Mathematics and Mathematical Physics, Vol. 69B, No. 4 (Oct.-Dec. 1965), pp. 335-336.
%H Mathias Lerch, <a href="https://eudml.org/doc/158206">Zur Theorie des Fermatschen Quotienten (a^(p-1) - 1)/p == q(a)</a>, Mathematische Annalen, Vol. 60 (1905), pp. 471-490.
%H D. Mirimanoff, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k6294283c/f12.item">Sur le dernier théorème de Fermat</a>, C. R. Acad. Sci. Paris, Vol. 150 (1910), pp. 204-206. Revised as <a href="https://eudml.org/doc/149351">Sur le dernier théorème de Fermat</a>, Journal für die reine und angewandte Mathematik, Vol. 139 (1911), pp. 309-324.
%H Planet Math, <a href="http://planetmath.org/WieferichPrime">Wieferich Primes</a>.
%H Reese Scott and Robert Styer, <a href="https://doi.org/10.1016/j.jnt.2005.09.001">On the generalized Pillai equation +-a^x +-b^y = c</a>, Journal of Number Theory, Vol. 118, No. 2 (2006), pp. 236-265.
%t Select[Prime[Range[1000000]], PowerMod[3, # - 1, #^2] == 1 &] (* _Robert Price_, May 17 2019 *)
%o (PARI)
%o N=10^9; default(primelimit,N);
%o forprime(n=2,N,if(Mod(3,n^2)^(n-1)==1,print1(n,", ")));
%o \\ _Joerg Arndt_, May 01 2013
%o (Python)
%o from sympy import prime
%o from gmpy2 import powmod
%o 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
%Y Cf. A039951, A096082.
%Y 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).
%K nonn,hard,bref,more
%O 1,1
%A _N. J. A. Sloane_
%E Edited by _Max Alekseyev_, Oct 20 2010
%E Updated by _Max Alekseyev_, Jan 29 2012
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