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Near-Wieferich primes with abs(A) < 2.
4

%I #16 Apr 29 2022 17:29:29

%S 3,5,7,71,379,1093,2659,3511,110057537,47004625957,76843523891

%N Near-Wieferich primes with abs(A) < 2.

%C Primes p such that A258367(i) < 2, where i is the index of p in A000040.

%C Subsequence of A246568.

%C Wieferich primes together with the "closest" near-misses possible that are not actually Wieferich.

%C Countless such sequences with larger bounds on the value of abs(A) are possible. This is one of the few that I believe should be in the OEIS.

%C The corresponding sequence of A-values is 1, 1, 1, 1, -1, 0, -1, 0, -1, 1, 1.

%C I checked the range 3 <= p <= 47004625957 with PARI. 76843523891 is from Crandall, Dilcher, Pomerance, 1997.

%C There are no near-Wieferich primes with abs(A) < 2 in the range 4*10^12 to 1.25*10^15 (cf. Knauer, Richstein, 2005).

%C There are no near-Wieferich primes with abs(A) < 2 in the range 1*10^15 to 3*10^15 per information I received from Mark Rodenkirch in 2010.

%C There are no near-Wieferich primes with abs(A) < 2 in the range 3*10^15 to ~6*10^17 (cf. Goetz, cf. Reggie, cf. Fries).

%C As of Apr 26 2022, a(12) > ~1.1*10^19 if it exists (cf. WW Statistics).

%C Heuristically, one would expect about 11 to 12 (3*log(log(10^19))) near-Wieferich primes with |A| <= 1 up to 10^19, a very close match to the actual number of 11 (cf. Crandall, Dilcher, Pomerance, 1997, p. 446).

%H R. Crandall, K. Dilcher and C. Pomerance, <a href="https://doi.org/10.1090/S0025-5718-97-00791-6">A search for Wieferich and Wilson primes</a>, Mathematics of Computation 66 (1997), 433-449.

%H U. Fries, <a href="https://u-g-f.de/PRPNet/findlist_adv.php?proj=WFS">PRPNet findlist for project WFS</a>

%H M. Goetz, <a href="http://www.primegrid.com/forum_thread.php?id=9455">WW by the numbers</a>, PrimeGrid forum.

%H J. Knauer and J. Richstein, <a href="https://doi.org/10.1090/S0025-5718-05-01723-0">The continuing search for Wieferich primes</a>, Mathematics of Computation 74 (2005), 1559-1563.

%H J. S. Nielsen, <a href="http://www.primegrid.com/forum_thread.php?id=9444&amp;nowrap=true#145728">Check a WW find with PARI/GP</a>, PrimeGrid forum.

%H Reggie, <a href="http://www.primegrid.com/forum_thread.php?id=9479&amp;nowrap=true#146701">Which near-finds are shown?</a>, PrimeGrid forum (see Message 146701).

%H PrimeGrid, <a href="http://www.primegrid.com/stats_ww.php">WW Statistics</a>

%e The prime p = 110057537 satisfies 2^((p-1)/2) == +1 -p (mod p^2) and is therefore in the sequence.

%o (PARI) a258367(n) = abs(centerlift(Mod(2, n^2)^((n-1)/2))\/n) \\ after _Charles R Greathouse IV_ in A258367

%o forprime(p=3, , if(a258367(p) < 2, print1(p, ", ")))

%o (PARI) \\ see Nielsen link for code to check the A-value and +-1-type of a prime

%Y Cf. A000040, A001220, A246568, A258367.

%K nonn,hard,more

%O 1,1

%A _Felix Fröhlich_, Apr 26 2022