OFFSET
1,1
COMMENTS
99 is the smallest base where more than two Wieferich pseudoprimes exist that can be found quickly (and hence, at this time, the "earliest" sequence of Wieferich pseudoprimes that should perhaps be in the OEIS).
Any other term is larger than 1.479*10^14 (cf. Fischer). This follows from the fact that the prime factors of a base-b Wieferich pseudoprime are base-b Wieferich primes (a necessary, but not sufficient condition for a composite to be a base-b Wieferich pseudoprime). The base-99 Wieferich primes p less than 1.479*10^14 are 5, 7, 13, 19, 83. All of them satisfy the congruence 99^(p-1) == 1 (mod p^k) for k = 2, but not for k = 3 (and hence for no k > 3). If any of them were repeated factors of a Wieferich pseudoprime, they would have to satisfy the congruence for k > 2. The squarefree products of these five primes can be excluded by a quick brute-force search, as all those products are extremely small (less than 10^6).
Note the following curious observation: The base-33 representations of a(1), a(2) and a(3) are [1, 2], [1, 32] and [1, 19, 13], respectively. The digit-sums of the decimal values of these digits are 3, 33, 33, respectively. Is it just a coincidence that all of these values consist only of 3s?
LINKS
PROG
(PARI) forcomposite(c=1, , if(Mod(99, c^2)^(c-1)==1, print1(c, ", ")))
CROSSREFS
KEYWORD
nonn,hard,bref,more
AUTHOR
Felix Fröhlich, Apr 28 2022
STATUS
approved