Does the string of prime numbers contain all natural numbers?

Does the string of prime numbers 2357111317 contain every natural number as its sub-string?

Answer

It follows from Dirichlet’s Theorem.

If d is the number we want to find, define s=10d+1. By definition, gcd and s contains the digits of d.

Dirichlet’s Theorem’s implies there’s a prime of the form p:=s+k \times 10^n where 10^n is chosen so that it has as many zeroes as digits of s. The digits of d appear in the digits of p, and thus in the given string of primes.

Attribution
Source : Link , Question Author : Buddha , Answer Author : Rebecca J. Stones

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