Wieferich primes in base $47$

Question

Wieferich primes are defined as prime numbers $p$ such that $p^2$ divides $2^{p − 1} − 1$. While reading about such primes, I came upon the following curious conjecture on the Wikipedia page of "unsolved problems in number theory":

Are there any Wieferich primes in base $47$?

Since no explanation is given for this strange question, I find myself puzzled by the importance of the number $47$ within this context. What role does this base in particular have in the context of Wieferich primes and why would it be important to solve this problem in particular, instead of another number base?

Answer 1

After some research on the internet, it indeed seems that Peter was right in the comment section, and $47$ is the smallest base for which no Wieferich primes are known.

Answer 2

Weiferich primes exist only in base 2. He wrote his paper in 1908, but the question has been around since the time of Euler. Dickson's history of mathematics devotes 12 pages to this question, but affords Weiferich only five lines in a paragraph on the nineth page. It is misleading to use this term.

Weiferich did not discover sevenites, but he noted that a particular solution to fermat's last proposition exists only for binary sevenites.

There is no particular reason for 47 not to have any particular sevenites. A similar situation exists with there being no known iso-sevenites for 3, (the Fibonacci series) which means that no instance of where if $p \mid F_n$ then also $p^2 \mid F_n$. However, the octagon-series and the Heron series, which correspond to isobases 6 and 4, do have sevenites.

Thus, in the series of Heron triangles, (triangles of sides e-1, e, e+1 and integer area), if 103 divides a side, so does $103^2$.

47 has 2 as a sevenite (or 'weiferich prime'). The two-place period of 2 supposes only 8, as can be seen in 11, 13 and 45. Here 32 divides 47^2-1, and thus it has two as a sevenite.

EDIT:

The table of 'sevenites' for particular bases, up to b=14400 and p=2000000, do not produce a list of primes longer than 80 digits, except in one or two cases. The number is quite small. Sort of in the $\sum 1/p$ range. There are a good scattering of unfilled rows, 47 is the first.

In the early versions of the tables I produced 6 was the first unfilled row. But then 61661 came along.