Prime digit primes.

The last hours I was working at new ideas for problems. Then I was wondering about a special kind of prime numbers.

Prime numbers where every digit itself is a prime numbers. In a nutshell I am interested in prime numbers, which only consists of the digits 2,3,5 or 7.

When the numbers grow bigger and bigger, then there are even less such numbers (which seems clearly, because every number with a 1,4,6,8 or 9 at the beginning can't be such a number (and 5/9 of the numbers between 1000 and 10000 start with 1,4,6,8 or 9, and there are even less primes)).

Is there a biggest prime number, which satisfy the condition? And is it possible to proof this?

Thank's for trying.

#NumberTheory

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$