Conjecture:
For any positive integer $n$, there exists a prime whose first digits are the digits of $n$.
Proof:
The infinite set of integers starting with the digits of integer $n$ is distributed as $1$ sequence of consecutive integers for each power of $10$. For example, the integers starting with the same digits as $2$ are those between $20$ and $30$, between $200$ and $300$, between $2000$ and $3000$ $…$ Now let's say $n=1245$, and let's look at much larger integers than $n$. For example, all integers $x$ in the following interval starts with digits $1245$: $$ 12450000000000000000000000000 ≤ x <12460000000000000000000000000 $$
A range of this magnitude is proven to contain at least $1$ prime number, so whether $n$ is a single digit or a million digits long, we only need to add the proper number of zeros to reach an interval where having no prime is impossible.
Questions:
- Is my proof valid ? Did I miss something ?
- Is there other existing proofs of that ?
Of course any suggestions/edits to make it more clear or professional are welcome, I'm still a newbie at proof writing!