|
| |
|
|
A306885
|
|
Minimal near-Wieferich A-value (absolute) for all primes in the interval [10^n, 10^(n+1)].
|
|
1
|
|
|
|
1, 1, 1, 0, 2, 5, 6, 17, 1, 6, 1, 19, 6, 2, 2, 10, 26
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
In other words, minimal value of abs(A) among A-values of all (n+1)-digit primes such that 2^{(p-1)/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
|
|
|
|
EXAMPLE
|
For n = 1: The A-values 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)^((n-1)/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
|
|
|
|
KEYWORD
|
nonn,hard,more
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
