# Are there infinitely many unique-period prime numbers?

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.

• Were you expecting that OEIS would list infinitely many unique primes? Dec 1 '21 at 2:00
• I can't contribute to any answer, but I wonder: why repeating decimal (which is not a prime itself)? Why not other number bases? Dec 1 '21 at 10:55
• @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). Dec 1 '21 at 10:58
• @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. Dec 1 '21 at 12:56
• Wolfram.mathworld.com ought to be chastised for perpetuating such a blatantly clickbaity and misleading name. Dec 1 '21 at 13:47

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.
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.