We know the primes are rich in arithmetic progressions (and in fact, any set with positive “upper density” in the primes is also rich in arithmetic progressions

We know the primes are rich in arithmetic progressions (and in fact, any set with positive “upper density” in the primes is also rich in arithmetic progressions). So we do know that there are 100,000,000,000! primes that are equidistant from one another, which is neat.