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A246568 Near-Wieferich primes (primes p satisfying 2^((p-1)/2) == +-1 + A*p (mod p^2)) with |A| < 10. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

Felix Fröhlich, Table of n, a(n) for n = 1..48 (all terms up to 3*10^15)

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

L. Vogel et al., PRPNet (the PRPClient package includes the program  "wwww")

PROG

(PARI) forprime(p=3, 3e15, for(a=-9, 9, if(Mod(2, p^2)^((p-1)/2)==1+a*p, print1(p, ", "); break({1})); if(Mod(2, p^2)^((p-1)/2)==-1+a*p, print1(p, ", "); break({1}))))

(wwww) -t Wieferich -p 2 -P 3e15 -s 10

CROSSREFS

Cf. A001220, A195988, A241014, A244801.

Sequence in context: A139758 A306084 A060770 * A120334 A000978 A128925

Adjacent sequences:  A246565 A246566 A246567 * A246569 A246570 A246571

KEYWORD

nonn,hard

AUTHOR

Felix Fröhlich, Aug 30 2014

STATUS

approved

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Last modified June 20 17:27 EDT 2019. Contains 324234 sequences. (Running on oeis4.)