To prove the following fact: given any sequence of digits in any base, eg 314159265358979323 base 10, there are infinitely many primes that start with these digits,eg when expressed in decimal they start with 314159265358979323. I think using a prime number theorem will be helpful . But I am still not getting a point of how to start the proof.
Proof that there are infinitely many prime numbers starting with a given digit string
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prime-numbers
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1The "Algorithms" in the title doesn't seem appropriate. – 2011-08-30
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0@Sri: Any suggestions for a better title? You could propose one by clicking edit, and users with sufficient rep can approve it if it's hunky-dory. – 2011-08-30
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0Note that "$n$ starts with $314159265358979323$" is equivalent to "there exists $k \in \mathbb N$ such that $314159265358979323 \cdot 10^k \le n < 314159265358979324 \cdot 10^k$". Can you show that the union of these intervals for all $k$ must contain infinitely many primes? – 2011-08-30
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3I am a little suspicious that two people asked the same question less than an hour apart. Can you give the source of the question? – 2011-08-30
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0@Sri: I certainly didn't read it as curt; I was merely saying that you can always suggest better titles (via the edit button) for questions whose titles are a bit short in the quality department... :) – 2011-08-30
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0Related: https://math.stackexchange.com/questions/2440482 – 2018-11-28