Base-a Wieferich Primes

Before delving into what base-\(a\) Wieferich primes are, we need to be clear about what is meant by Wieferich prime. Interestingly, only two are known.  Here is a quote from Wikipedia:

In number theory, a Wieferich prime is a prime number \(p\) such that \(p^2\) divides \(2^{p − 1} − 1\), therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides \(2^{p − 1}− 1\). Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.

Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the \(abc\) conjecture.

As of March 2021, the only known Wieferich primes are 1093 and 3511 (sequence A001220 in the OEIS).

If we relax the condition that the base has to be 2 then we open the door to what are called base-\(a\) Wieferich primes. Let's make \(a=3\) and ask the question what primes satisfy:$$3^{p-1} \equiv 1 \pmod{p^2} \text{ or } 3^{p-1} -1\equiv 0 \pmod{p^2}$$A little testing will show that the smallest value of \(p\) to satisfy this condition is 11. The next prime to satisfy is 1006003 and no prime has been found beyond that as yet.

OEIS A039951 records the smallest solutions for \(a^{p-1} \equiv 1 \pmod{p^2} \) for various values of \(a\). The first 46 values are:

2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, 2693, 2, 29, 29131, 1093, 2, 5, 3, 281, 2, 13, 13, 5, 2, 3, 11, 3, 2, 7, 7, 5, 2, 46145917691, 3, 66161, 2, 17, 8039, 11, 2, 23, 5, 3, 2, 3

No primes have been found that satisfy 47, 72, 186, 187, 200, 203, 222, 231, 304, 311, 335, 355, 435, 454, 546, 554, 610, 639, 662, 760, 772, 798, 808, 812, 858, 860, 871, 983, 986, ...  although it is conjectured that there are infinitely many solutions for every \(a\). Table 1 shows Wieferich primes for various values of \(a\):

Table 1: source

I came across these primes when researching properties associated with my diurnal age of 26645.  It turns out that this number is a member of OEIS A344827:

A344827

\(s_n\) is the smallest \(a \gt 1\) such that prime(\(n\)), prime(\(n\)+1) and prime(\(n\)+2) are all base-\(a\) Wieferich primes.

The members of the sequence from \(n=1 \dots 8\) are 449, 226, 1207, 606, 3469, 653, 5649 and 26645. Now prime(8)=19 and thus we have 26645 as the smallest value of \(a\). Thus we have:$$26645^{19-1} \equiv 1 \pmod{19^2} \\26645^{23 -1} \equiv 1 \pmod{23^2} \\26645^{29-1}  \equiv 1 \pmod{29^2}$$