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 A090968 Primes p such that p^2 divides 19^(p-1) - 1. 18
 3, 7, 13, 43, 137, 63061489 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. REFERENCES 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. [Harvey P. Dale, Oct 17 2011] LINKS Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5. C. Caldwell, Fermat quotient W. Keller and J. Richstein Fermat quotients q_p(a) that are divisible by p MATHEMATICA 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}] Select[Prime[Range[4*10^6]], PowerMod[19, #-1, #^2]==1&] (* Harvey P. Dale, Nov 08 2017 *) CROSSREFS Cf. A001220, A014127, A123692, A123693, A128667, A128668, A128669, A039951, A096082 Sequence in context: A174241 A289556 A086208 * A020641 A062736 A257716 Adjacent sequences:  A090965 A090966 A090967 * A090969 A090970 A090971 KEYWORD nonn,hard,more AUTHOR Robert G. Wilson v, Feb 27 2004 STATUS approved

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Last modified July 26 20:32 EDT 2021. Contains 346295 sequences. (Running on oeis4.)