A satanic prime is a prime number with $666$ in the decimal representation.
The smallest satanic prime is $6661$.
Prove that there are infinitely many satanic primes.
I used Dirichlet's theorem for the progression $10000n+6661$ and it is done.
I'm interested in solutions without Dirichlet's theorem.
Best Answer
Consider the set $S$ of all numbers without 666 in their base 10 expression. Here's a fun fact: the sum $\sum_{s\in S} \frac{1}{s}$ converges. It's actually pretty easy to prove, so I'll leave it as an exercise (or google "Kempner series").
On the other hand, a famous result of Euler says the sum of the reciprocals of the prime numbers diverges.