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

3, 487, 56598313

Offset: 1

Author

Helmut Richter, Dec 11 1999

Keywords

• bref
• hard
• nonn
• nice
• more

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

From Jianing Song, Jun 21 2025: (Start)

The ring of integers of Q(10^(1/k)) is Z[10^(1/k)] if and only if k does not have a prime factor in this sequence. See Theorem 5.3 of the paper of Keith Conrad. For example, we have:

(1 + 10^(1/3) + 10^(2/3))/3 is an algebraic integer, but it is not in Z[10^(1/3)];

(1 + 10^(486/487) + 10^(2*486/487) + ... + 10^(486*486/487))/487 is an algebraic integer, but it is not in Z[10^(1/487)];

(1 + 10^(56598312/56598313) + 10^(2*56598312/56598313) + ... + 10^(56598312*56598312/56598313))/56598313 is an algebraic integer, but it is not in Z[10^(1/56598313)]. (End)

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

• Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
• Keith Conrad, The ring of integers in a radical extension.
• 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.
• Siqiong Yao and Akira Toyohara, The length of the repeating decimal, arXiv:2507.01295 [math.NT], 2025. See p. 19.
• Samuel Yates, The Mystique of Repunits, Math. Mag. 51 (1978), 22-28.

Crossrefs

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

Cf. A265012, A049084, A000040.

Programs

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

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 *)

PARI

lista(nn) = forprime(p=2, nn, if (Mod(10, p^2)^(p-1)==1, print1(p, ", "))); \\ Michel Marcus, Aug 16 2015