There are many primes which have digits only from the set of one digit primes $\mathcal S_p = \{2,3,5,7\}$, but can we prove that there exists at least one $p$ for each $n$: $10^n \leq p \leq10^{n+1}$, $n\in \mathbb N$ with digits only in $\mathcal S_p$?

some first few examples $$7, 23, 257, 2357, \cdots$$

It feels like something easy to verify for many $n$ but difficult to prove for all.

EDIT This is not the same question as the duplicate marker at all. It asks about primes of all sizes, not only three digit and only asks about solely being built up by single-digit primes.

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    $\begingroup$ Possible duplicate of Twice prime number? $\endgroup$ – Paolo Leonetti Jun 29 '17 at 9:21
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    $\begingroup$ There are $4^n$ candidate numbers with $n$ digits. The "probability" of being prime is $\sim\frac 1n$, so it is quite feasible to assume such primes exist with all digit lengths once we make it through the small $n$ $\endgroup$ – Hagen von Eitzen Jun 29 '17 at 9:25
  • $\begingroup$ @PaoloLeonetti I agree that the question is similar, but it is not a duplicate. The OP asks if one can prove that there exists at least one $p$ for each $n$: $10^n \lt p \lt 10^{n+1}, n \in \mathbb{N}$, while the original question only asks if a three-digit number with this property exists (the single, accepted answer reflects this). $\endgroup$ – jvdhooft Jun 29 '17 at 9:25
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    $\begingroup$ The following is the number of primes with n digits in the set $\{2,3,5,7\}$ with $2\le n\le 12$:$$4,\ 15,\ 38,\ 128,\ 389,\ 1325,\ 4643,\ 16623,\ 59241,\ 214432,\ 781471$$ $\endgroup$ – Julián Aguirre Jun 29 '17 at 11:00
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    $\begingroup$ @JulianAguirre: Wow that seems a fast increasing function. Maybe a more interesting question would be "how many percent of the primes will be of this kind for different $n$". $\endgroup$ – mathreadler Jun 29 '17 at 11:22

In 2016 James Maynard proved that there are infinite number of primes without using one digit and he got the EMS award for it, if your conjecture was correct then that would be a stronger statement then his.

On the other hand it would probably be easier to prove that there are only finite repunit primes (primes which have only $1$'s in their decimal representation) then that there are only finite primes with having $2,3,5,7$ as their only digits.

According to Prime number theorem and the OEIS sequence it seems rather unlikely that if there are infinite number of such primes that there isn't at least one with $n$ digits


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