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Primes p such that 10^(p-1) == 1 (mod p^2).
20

%I #54 Feb 29 2020 06:46:31

%S 3,487,56598313

%N Primes p such that 10^(p-1) == 1 (mod p^2).

%C Primes p such that the decimal fraction 1/p has same period length as 1/p^2, i.e., the multiplicative order of 10 modulo p is the same as the multiplicative order of 10 modulo p^2. [extended by _Felix Fröhlich_, Feb 05 2017]

%C No further terms below 1.172*10^14 (as of Feb 2020, cf. Fischer's table).

%C 56598313 was announced in the paper by Brillhart et al. - _Helmut Richter_, May 17 2004

%C A265012(A049084(a(n))) = 1. - _Reinhard Zumkeller_, Nov 30 2015

%D J. Brillhart, J. Tonascia, and P. Weinberger, On the Fermat quotient, pp. 213-222 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.

%D Richard K. Guy, Unsolved Problems in Number Theory, Springer, 2004, A3.

%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 Richard Fischer, <a href="http://www.fermatquotient.com/FermatQuotienten/">Fermat quotients B^(P-1) == 1 (mod P^2)</a>.

%H Wilfrid Keller and Jörg Richstein, <a href="https://doi.org/10.1090/S0025-5718-04-01666-7">Solutions of the congruence a^(p-1) == 1 (mod p^r)</a>, Math. Comp. 74 (2005), 927-936.

%H Peter L. Montgomery, <a href="https://doi.org/10.1090/S0025-5718-1993-1182246-5">New solutions of a^(p-1) == 1 (mod p^2)</a>, Math. Comp. 61 (1993), 361-363.

%H Math Overflow, <a href="http://mathoverflow.net/questions/27579/is-the-smallest-primitive-root-modulo-p-a-primitive-root-modulo-p2">Is the smallest primitive root modulo p a primitive root modulo p^2?</a>, Jun 09 2010.

%H Helmut Richter, <a href="http://hr.userweb.mwn.de/numb/period.html">The period length of the decimal expansion of a fraction</a>.

%H Helmut Richter, <a href="http://hr.userweb.mwn.de/numb/fact486.html">The Prime Factors Of 10^486-1</a>.

%H Samuel Yates, <a href="http://www.jstor.org/stable/2689643">The Mystique of Repunits</a>, Math. Mag. 51 (1978), 22-28.

%t A045616Q = PrimeQ@# && PowerMod[10, # - 1, #^2] == 1 &; Select[Range[1000000], A045616Q] (* _JungHwan Min_, Feb 04 2017 *)

%t Select[Prime[Range[34*10^5]],PowerMod[10,#-1,#^2]==1&] (* _Harvey P. Dale_, Apr 10 2018 *)

%o (PARI) lista(nn) = forprime(p=2, nn, if (Mod(10, p^2)^(p-1)==1, print1(p, ", "))); \\ _Michel Marcus_, Aug 16 2015

%o (Haskell)

%o import Math.NumberTheory.Moduli (powerMod)

%o a045616 n = a045616_list !! (n-1)

%o a045616_list = filter

%o (\p -> powerMod 10 (p - 1) (p ^ 2) == 1) a000040_list'

%o -- _Reinhard Zumkeller_, Nov 30 2015

%Y Cf. A001220, A014127, A123692, A212583, A123693, A111027, A128667, A234810, A242741, A128668, A244260, A090968, A242982, A128669, A039951.

%Y Cf. A265012, A049084, A000040.

%K bref,hard,nonn,nice,more

%O 1,1

%A _Helmut Richter_, Dec 11 1999