A385856 Near-Wieferich primes (primes p satisfying 2^p == 2 + A*p (mod p^2)) with |A| <= 10.

2, 3, 5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 47, 71, 157, 173, 211, 251, 263, 379, 383, 1093, 1097, 1699, 1753, 2633, 2659, 3373, 3511, 3593, 5501, 8089, 10691, 15823, 27967, 30577, 45827, 46477, 1437049, 1483597, 1897121, 2152849, 6266543, 52368101, 110057537, 126233057, 1683955849, 2001907169, 13211006161, 47004625957

Offset: 1

Author

A. Lamek, Jul 10 2025

Keywords

• nonn

Comments

Near-Wieferich primes: 2^p == 2 + A*p (mod p^2) for |A| <= 10. Extends the Wieferich condition by allowing small symmetric offsets.

See also A246568 for a related formulation involving the same congruence structure.

All values verified for p <= 5*10^10.

Examples

			p=11: 2^11 == 2048 == 2+(-1)*11 == -9 == 112 (mod 121), so A=-1.
p=5: 2^5 == 32 == 2+1*5 == 7 (mod 25), so A=1.
		

Links

• A. Lamek, User profile with notes on Wieferich-related structures

Crossrefs

Cf. A001220, A246568.

Programs

Mathematica

isokQ[p_] := Module[{A}, A = Quotient[PowerMod[2, p, p^2] - 2, p]; A <= 10 || p - A <= 10]

PARI

isok(p) = lift(Mod(2, p^2)^p-2+10*p) <= 20*p; \\ Michel Marcus, Jul 12 2025

Python

def is_a385856(p):
    A = ((pow(2, p, p*p)-2) // p) % p
    return (A<=10) or (p-A<=10)

Extensions

a(44)-a(46) from Michel Marcus, Jul 12 2025

a(48) from Jinyuan Wang, Jul 13 2025