A045616 Primes p such that 10^(p-1) == 1 (mod p^2).

3, 487, 56598313

Offset

1,1

Comments

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]

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

56598313 was announced in the paper by Brillhart et al. - Helmut Richter, May 17 2004

A265012(A049084(a(n))) = 1. - Reinhard Zumkeller, Nov 30 2015

References

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.

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

Links

Table of n, a(n) for n=1..3.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.

Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2).

Wilfrid Keller and Jörg Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.

Peter L. Montgomery, New solutions of a^(p-1) == 1 (mod p^2), Math. Comp. 61 (1993), 361-363.

Math Overflow, Is the smallest primitive root modulo p a primitive root modulo p^2?, Jun 09 2010.

Helmut Richter, The period length of the decimal expansion of a fraction.

Helmut Richter, The Prime Factors Of 10^486-1.

Samuel Yates, The Mystique of Repunits, Math. Mag. 51 (1978), 22-28.

Mathematica

A045616Q = PrimeQ@# && PowerMod[10, # - 1, #^2] == 1 &; Select[Range[1000000], A045616Q] (* JungHwan Min, Feb 04 2017 *)
Select[Prime[Range[34*10^5]], PowerMod[10, #-1, #^2]==1&] (* Harvey P. Dale, Apr 10 2018 *)

Prog

(PARI) lista(nn) = forprime(p=2, nn, if (Mod(10, p^2)^(p-1)==1, print1(p, ", "))); \\ Michel Marcus, Aug 16 2015
(Haskell)
import Math.NumberTheory.Moduli (powerMod)
a045616 n = a045616_list !! (n-1)
a045616_list = filter
               (\p -> powerMod 10 (p - 1) (p ^ 2) == 1) a000040_list'
-- Reinhard Zumkeller, Nov 30 2015

Crossrefs

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

Cf. A265012, A049084, A000040.

Sequence in context: A230029 A238447 A241977 * A198705 A198624 A198652

Adjacent sequences: A045613 A045614 A045615 * A045617 A045618 A045619

Keyword

bref,hard,nonn,nice,more

Author

Helmut Richter, Dec 11 1999

Status

approved