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A045616 Primes p such that 10^(p-1) == 1 (mod p^2). 13
3, 487, 56598313 (list; graph; refs; listen; history; text; internal format)



Primes p such that the decimal fraction 1/p has same period length as 1/p^2.

No further terms below 3.4*10^13 (cf. Fischer's table).

56598313 was announced in the paper by Brillhart et al. - Helmut Richter (richter(AT)lrz.de), May 17 2004

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


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.


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

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

P. L. Montgomery, New solutions of a^p-1 == 1 (mod p^2), Math. Comp., 61 (203), 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.


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


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


Cf. A001220, A014127, A123692, A123693, A111027, A039951.

Cf. A265012, A049084, A000040.

Sequence in context: A230029 A238447 A241977 * A198705 A198624 A198652

Adjacent sequences:  A045613 A045614 A045615 * A045617 A045618 A045619




Helmut Richter (richter(AT)lrz.de)



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Last modified February 12 08:47 EST 2016. Contains 268212 sequences.