For any $n \geq 1$, prove that there exists a prime $p$ with at least n of its digits equal to $0$. I don't even know how to start?? Any help(even a hint) would do. Thanks in advance!!
For any $n \geq 1$, prove that there exists a prime $p$ with at least n of its digits equal to $0$
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number-theory
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0@Cocopuffs:you re$a$lly simplified my problem(some others too).Thanks for the help. – 2012-06-23
2 Answers
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The series $\sum_{p \in \mathbb P} \frac{1}{p}$ of primes diverges (proof). Show that the series $\sum_{n \in A} \frac{1}{n}$ converges, where $A$ is the set of integers with at most $k$ zeroes (modify this proof). Therefore $\mathbb P \not \subset A$.
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1This seems to be the only solution that requires no technology beyond the 18th century! – 2012-06-23
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By Dirichlet's theorem on primes in arithmetic progressions there are infinitely many primes congruent to $1$ mod $10^{n+1}$. These primes have at least $n$ $0$'s in them.