

A259909


nth Wieferich prime to base prime(n), i.e., primes p such that p is the nth solution of the congruence (prime(n))^(p1) == 1 (mod p^2).


1




OFFSET

1,1


COMMENTS

Main diagonal of table T(b, p) of Wieferich primes p to prime bases b (that table is not yet in the OEIS as a sequence).
a(4), if it exists, corresponds to A123693(4) and is larger than 9.7*10^14 (cf. Dorais, Klyve, 2011).
a(5), if it exists, corresponds to the 5th base11 Wieferich prime and is larger than approximately 5.9*10^13 (cf. Fischer).
a(6), if it exists, corresponds to A128667(6) and is larger than approximately 5.9*10^13 (cf. Fischer).


REFERENCES

W. Keller, Prime solutions p of a^p1 = 1 (mod p2) for prime bases a, Abstracts Amer. Math. Soc., 19 (1998), 394.


LINKS

Table of n, a(n) for n=1..3.
M. Aaltonen and K. Inkeri, Catalan's equation x^p  y^q and related congruences, Mathematics of Computation, Vol. 56 No. 193 (1991), 359370.
F. G. Dorais and D. Klyve, A Wieferich prime search up to p < 6.7*10^15, J. Integer Seq. 14 (2011), Art. 11.9.2, 114.
R. Fischer, Thema: Fermatquotient B^(P1) == 1 (mod P^2)
W. Keller and J. Richstein, Fermat quotients q_p(a) that are divisible by p (Cached copy at the Wayback Machine).
K. E. Kloss, Some NumberTheoretic Calculations, J. Research of the National Bureau of StandardsB. Mathematics and Mathematical Physics, Vol. 69B, No. 4 (1965), 335336.


EXAMPLE

a(1) = A001220(1) = 1093.
a(2) = A014127(2) = 1006003.
a(3) = A123692(3) = 40487.


PROG

(PARI) a(n) = my(i=0, p=2); while(i < n, if(Mod(prime(n), p^2)^(p1)==1, i++; if(i==n, break({1}))); p=nextprime(p+1)); p


CROSSREFS

Cf. A001220, A014127, A123692, A123693, A174422, A178871.
Sequence in context: A307220 A091674 A022197 * A124122 A163561 A203807
Adjacent sequences: A259906 A259907 A259908 * A259910 A259911 A259912


KEYWORD

nonn,hard,bref,more


AUTHOR

Felix FrÃ¶hlich, Jul 07 2015


STATUS

approved



