A306885 Minimal near-Wieferich A-value (absolute) for all primes in the interval [10^n, 10^(n+1)].

1, 1, 1, 0, 2, 5, 6, 17, 1, 6, 1, 19, 6, 2, 2, 10, 26

Offset: 0

Author

Felix Fröhlich, Mar 15 2019

Keywords

• nonn
• hard
• more

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.

Examples

			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.
		

Links

• R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation 66 (1997), 433-449.
• J. Knauer and J. Richstein, The continuing search for Wieferich primes, Mathematics of Computation 74 (2005), 1559-1563.
• PrimeGrid, WW stats
• Sysadm@Nbg and PrimeGrid, PRPNet findlist for project WFS [Archived version at the Wayback Machine].

Crossrefs

Cf. A001220, A258367.

Programs

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

Extensions

a(15)-a(16) from Felix Fröhlich, Apr 03 2021