Is every natural number a prefix of a prime number?

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Proof that there are infinitely many prime numbers starting with a given digit string

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

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 — hmakholm left over Monica, Jun 08 '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 08 '12 at 16:18

Wouter Zeldenthuis · Answer 1 · 2012-06-08T16:31:59.643

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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.

edited Jun 08 '12 at 16:31

answered Jun 08 '12 at 16:18

Wouter Zeldenthuis

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This simply restates the question and is insubstantial. – Evgeni Sergeev Jun 12 '15 at 11:00

Is every natural number a prefix of a prime number?

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