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


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.

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

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


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

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

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

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


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

Sequence in context: A230029 A238447 A241977 * A198705 A198624 A198652

Adjacent sequences:  A045613 A045614 A045615 * A045617 A045618 A045619




Helmut Richter (richter(AT)lrz.de)


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



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Last modified December 18 10:27 EST 2014. Contains 252139 sequences.