

A306885


Minimal nearWieferich Avalue (absolute) for all primes in the interval [10^n, 10^(n+1)].


0



1, 1, 1, 0, 2, 5, 6, 17, 1, 6, 1, 19, 6, 2, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,5


COMMENTS

In other words, minimal value of abs(A) among Avalues of all (n+1)digit primes such that 2^{(p1)/2} == +1 + A*p (mod p^2).
a(n) = 0 indicates that at least one (n+1)digit Wieferich prime (A001220) exists. In particular, a(3) = 0, because the interval [10^3, 10^4] contains the two Wieferich primes 1093 and 3511.
Clearly, the number of 0 terms is infinite if and only if A001220 is infinite.
a(9)a(11) are from Crandall, Dilcher, Pomerance, 1997.
a(12) from data in Crandall, Dilcher, Pomerance, 1997 and Knauer, Richstein, 2005.
a(13)a(14) from Knauer, Richstein, 2005.
In Crandall, Dilcher, Pomerance, 1997, a heuristic argument is given that predicts the number of Wieferich primes below some bound x to be about log(log(x)). If that heuristic is accurate, then one could expect the next 0 to occur at n with 9 <= n <= 24.


LINKS

Table of n, a(n) for n=0..14.
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation 66 (1997), 433449.
J. Knauer and J. Richstein, The continuing search for Wieferich primes, Mathematics of Computation 74 (2005), 15591563.


EXAMPLE

For n = 1: The Avalues for the primes in the interval [10^1, 10^2] are 3, 5, 2, 8, 3, 14, 3, 18, 9, 9, 22, 18, 4, 18, 5, 1, 28, 30, 24, 3, 20. The smallest of these, by absolute value, is 1, so a(1) = 1.


PROG

(PARI) a258367(n) = abs(centerlift(Mod(2, n^2)^((n1)/2))\/n)
a(n) = my(minm=nextprime(10^n)); forprime(p=10^n, 10^(n+1), if(p!=2, if(a258367(p) < minm, minm=a258367(p)))); minm


CROSSREFS

Cf. A001220, A258367.
Sequence in context: A101325 A042980 A048290 * A029939 A082198 A098871
Adjacent sequences: A306882 A306883 A306884 * A306886 A306887 A306888


KEYWORD

nonn,hard,more


AUTHOR

Felix FrÃ¶hlich, Mar 15 2019


STATUS

approved



