# Is every natural number a prefix of a prime number? [duplicate]

Let n be the representation of a natural number in a non-unary base. Is it a prefix of the representation of a prime number over the same base?

For example: in decimal, the answer for 10 is yes, because 103 is prime. Is this true for every number?

EDIT: As Henning Makholm has pointed out, this question has been asked before: Proof that there are infinitely many prime numbers starting with a given digit string

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## marked as duplicate by Henning Makholm, Rahul, Steven Stadnicki, Jonas Meyer, Martin SleziakJun 8 '12 at 18:39

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Not only is there always such a prime -- there are infinitely many, as demonstrated in the question Proof that there are infinitely many prime numbers starting with a given digit string – Henning Makholm Jun 8 '12 at 16:17
I think your question is more complicated than it looks like. For instance, if you work in base $7$, the number $7$ is written $10$, so you're looking for a number of the form $10****$ in base $7$ that is prime. – Patrick Da Silva Jun 8 '12 at 16:18
Thank you, that was exactly what I was looking for. – Boris Trayvas Jun 8 '12 at 16:43

## 1 Answer

Yes. You need only to use basic results about the distributions of primes to guarantee that, for example, a prime number must exist between 100 and 109, or 1000-1099, etc. You should be able to easily generalize this.

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This simply restates the question and is insubstantial. – Evgeni Sergeev Jun 12 at 11:00