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A090968 Primes p such that p^2 divides 19^(p-1) - 1. 8
3, 7, 13, 43, 137, 63061489 (list; graph; refs; listen; history; text; internal format)



Primes p such that p divides the Fermat quotient of p (with base 19). The Fermat quotient of p with base a denotes the integer q_p(a) = ( a^(p-1) - 1) / p, where p is a prime which does not divide the integer a. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 20 2005

No further terms up to 3.127*10^13.


J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 43, p. 17, Ellipses, Paris 2008.

Paulo Ribenboim, "The Little Book Of Big Primes," Springer-Verlag, NY 1991, page 170.

Roozbeh Hazrat, Mathematica:A Problem-Centered Approach, Springer 2010, pp. 39, 171


Table of n, a(n) for n=1..6.

C. Caldwell, Fermat quotient

W. Keller and J. Richstein Fermat quotients q_p(a) that are divisible by p


NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ p = NextPrim[p]; If[PowerMod[19, p - 1, p^2] == 1, Print[p]], {n, 1, 2*10^8}]


Cf. A001220, A014127, A123692, A123693, A128667, A128668, A128669, A039951, A096082

Sequence in context: A191974 A174241 A086208 * A020641 A062736 A103564

Adjacent sequences:  A090965 A090966 A090967 * A090969 A090970 A090971




Robert G. Wilson v, Feb 27 2004


Third reference added by Harvey P. Dale, Oct. 17 2011



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Last modified December 22 22:37 EST 2014. Contains 252372 sequences.