On Convergent Sequences and Prime Numbers
Ever since Euclid first proved that there are infinitely many prime numbers, mathematicians have found ever more creative ways to prove the same result, and also various stronger theorems that imply it. Dirichlet’s Theorem, for example, states that if\(m\) and \(n\) are relatively prime integers, then there are infinitely many prime numbers of the form \(mk + n\) for some integer \(k\). It is also known that the sum of the reciprocals of the prime numbers diverges, that the sum
\[\displaystyle \sum_{\substack{p \leq n \\ p \text{ prime}}} \frac{1}{p} \sim \log(\log(n))\]
and that the number of prime numbers less than \(n\) is asymptotically equal to \(\displaystyle \frac{n}{\log(n)}\). In this blog post, we will continue this proud tradition by proving that there are infinitely many prime numbers which have your phone number somewhere in their digits, and which simultaneously have a prime number of digits.
To do so, we will look at the convergence of two different sums: that of the reciprocals of the primes with a prime number of digits, and that of the reciprocals of the natural numbers which do not contain your phone number amongst their digits.…