A278611 Bases b > 1 that set a new record for the size of the smallest base-b Wieferich prime.

2, 6, 34

Offset: 1

Author

Felix Fröhlich, Nov 23 2016

Keywords

• nonn
• hard
• more
• bref

Comments

Numbers n such that A039951(n) reaches a new record value.

a(1) = 2. Thereafter smallest number x that occurs later in column 1 of A244249 than any y with 1 < y < x.

Let b(n) be the sequence of corresponding smallest Wieferich primes. b(1)-b(3) are 1093, 66161 and 46145917691, respectively (cf. A307220).

No term is a perfect power, since then its smallest Wieferich prime is at most the size of the smallest Wieferich prime of the base that is raised to a power.

a(4) is either 47, 72 or 139, depending on which of those bases is the smallest where any Wieferich prime exists. The smallest base-139 Wieferich prime is 1822333408543 and any Wieferich primes in bases 47 and 72 are larger than 1.07*10^14 (cf. Fischer).

Examples

			a(2) = 6, since the smallest base-6 Wieferich prime is 66161 and that prime is the second term with a record value in A039951.
		

Links

• R. Fischer, Thema: Fermatquotient B^(P-1) == 1 (mod P^2)

Crossrefs

Cf. A039951, A244249, A307220.

Programs

PARI

smallest_w_prime(n) = forprime(p=1, , if(Mod(n, p^2)^(p-1)==1, return(p)))
my(r=0, b=2); while(1, if(smallest_w_prime(b) > r, print1(b, ", "); r=smallest_w_prime(b)); b++)