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
Among generalized Wieferich pairs (p, b) satisfying b^(p-1) == 1 (mod p^2) with 1 < b < p, ordered by increasing p, the base b = 19 is the second base that appears for two different primes, namely p = 43 and p = 137 (i.e., b = 19 is the first base to appear twice in G. Helms's table). The first base is b = 53. - William Hu, Jul 01 2026
No further terms up to 3.127*10^13.
Primes p such that binomial(19*p^k,p^k) == 19^p mod p^2 for some (or equivalently all) nonnegative integer k. - Chai Wah Wu, Jul 01 2026
REFERENCES
Roozbeh Hazrat, Mathematica: A Problem-Centered Approach, Springer 2010, pp. 39, 171.
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.
LINKS
Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
C. Caldwell, Fermat quotient
G. Helms, Tables of generalized Wieferich primes
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}]
(* Alternative: *)
Select[Prime[Range[4*10^6]], PowerMod[19, #-1, #^2]==1&] (* Harvey P. Dale, Nov 08 2017 *)
CROSSREFS
KEYWORD
nonn,hard,more,changed
AUTHOR
Robert G. Wilson v, Feb 27 2004
STATUS
approved