A353141 Near-Wieferich primes with abs(A) < 2.

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

Offset: 1

Author

Felix Fröhlich, Apr 26 2022

Keywords

• nonn
• hard
• more

Comments

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

Subsequence of A246568.

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

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.

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

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

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).

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.

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).

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

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).

Examples

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

Links

• R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation 66 (1997), 433-449.
• U. Fries, PRPNet findlist for project WFS
• M. Goetz, WW by the numbers, PrimeGrid forum.
• J. Knauer and J. Richstein, The continuing search for Wieferich primes, Mathematics of Computation 74 (2005), 1559-1563.
• J. S. Nielsen, Check a WW find with PARI/GP, PrimeGrid forum.
• Reggie, Which near-finds are shown?, PrimeGrid forum (see Message 146701).
• PrimeGrid, WW Statistics

Crossrefs

Cf. A000040, A001220, A246568, A258367.

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) < 2, print1(p, ", ")))

PARI

\\ see Nielsen link for code to check the A-value and +-1-type of a prime