A246568 Near-Wieferich primes (primes p satisfying 2^((p-1)/2) == +-1 + A*p (mod p^2)) with |A| < 10.

3, 5, 7, 11, 13, 17, 19, 23, 31, 41, 43, 59, 67, 71, 89, 127, 251, 379, 569, 571, 1093, 1427, 1451, 1733, 2633, 2659, 2903, 3511, 13463, 15329, 15823, 26107, 60631, 546097, 2549177, 110057537, 165322639, 209227901, 671499313, 867457663, 3520624567

Offset: 1

Author

Felix Fröhlich, Aug 30 2014

Keywords

• nonn
• hard

Comments

The data section gives all terms up to 10^10. There are eight more terms up to 3*10^15 (see b-file).

A is essentially (A007663(n) modulo A000040(n))/2 (see Crandall et al. (1997), p. 437). The choice of the bound for A is rather arbitrary and selecting a larger A will result in more terms in a specific interval. For any p there exist two values of A whose sum is p, except when p is in A001220, in which case A = 0.

Links

• Jeppe Stig Nielsen, Table of n, a(n) for n = 1..50 (terms n= 1..49 by Felix Fröhlich)
• R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp. Vol. 66, Num. 217 (1997), 433-449.
• J. Knauer and J. Richstein, The continuing search for Wieferich primes, Math. Comp. Vol. 74, Num. 251 (2005), 1559-1563.
• PrimeGrid, Wieferich & near Wieferich Primes p < 11e15
• PrimeGrid, WW Statistics

Crossrefs

Cf. A001220, A195988, A241014, A244801.

Programs

PARI

a258367(n) = abs(centerlift(Mod(2, n^2)^((n-1)/2))\/n) \\ after Charles R Greathouse IV in A258367
forprime(p=3, , if(a258367(p) < 10, print1(p, ", "))) \\ Felix Fröhlich, Apr 26 2022