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Wolfram.mathworld.com defines a unique prime in the following way:

"Following Yates (1980), a prime $p$ such that $\frac{1}{p}$ is a repeating decimal with decimal period shared with no other prime is called a unique prime. For example, $3$, $11$, $37$, and $101$ are unique primes."

OEIS has unique-period primes, and the list they give is finite. My question is whether we know if there are infinitely many unique primes, or if there is an easy way to prove that there are finitely many.

Quick clarification: I know that there are infinitely many primes already and how to prove that. I am particularly focused on unique primes.

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    $\begingroup$ Were you expecting that OEIS would list infinitely many unique primes? $\endgroup$
    – tparker
    Dec 1 '21 at 2:00
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    $\begingroup$ I can't contribute to any answer, but I wonder: why repeating decimal (which is not a prime itself)? Why not other number bases? $\endgroup$
    – nigel222
    Dec 1 '21 at 10:55
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    $\begingroup$ @tparker Of course not. The reason I mentioned is because oftentimes they will indicate that the sequence is infinite even when they only listen finitely many (and here they did not). $\endgroup$ Dec 1 '21 at 10:58
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    $\begingroup$ @nigel222 Because as a society we have agreed to use the decimal system, hence a lot of questions we consider are decimal-centric. There is no reason why we couldn't consider same questions for other bases though. $\endgroup$
    – Wojowu
    Dec 1 '21 at 12:56
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    $\begingroup$ Wolfram.mathworld.com ought to be chastised for perpetuating such a blatantly clickbaity and misleading name. $\endgroup$ Dec 1 '21 at 13:47
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Quoting from the 1998 paper "Unique-Period Primes" by Caldwell and Dubner,

Other than showing where to search for new unique-primes, we have not even addressed the most basic of questions about these primes: are there infinitely many of them? Are there infinitely many repunit primes? Are there infinitely $n$ such that $\Phi_n(10)$ is a power (greater than one) of a prime? We join others in conjecturing that the answers are yes, yes and no; but we are unable to prove any of these conjectures.

If there's any update, Caldwell has not cited it at the Prime Glossary entry.

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OEIS gives 16 in your link and 46 in another list with the largest shown being something like $(10^{1132}+1)/(10^4+1)$. It looks as if the list contains all known repunit primes (those just with digit $1$).

The problem is that candidates get too big to check whether they are primes. Suppose I told you that I thought $(10^{8177207}-1)/9$ was probably a unique prime (as I do), then you would have no practical way of checking.

There is currently no proof either way, but it is a plausible conjecture that there is an infinite number of them.

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