#infinite sum

Evelyn Lamb finds both aesthetic and intellectual value from @_primes_, a Twitterbot that tweets prime numbers.

  The harmonic series is the infinite sum 1+1/2+1/3+1/4+…, the sum of the reciprocals of every positive integer. Even though the terms get closer and closer to 0, the series goes to infinity, meaning it eventually gets bigger than any number you can throw at it. (The question is fun to think about, so I won’t spoil it for you, but Wikipedia has an explanation.) ...

One ingredient in making sense of this mystery is a puzzling fact: "most" large numbers have a 9 in them, so eventually you're throwing out an awful lot of the numbers from the harmonic series. This fact seems very strange to me because the intuition I have about numbers, primarily based on the one- and two-digit numbers we encounter the most, fails me.

Last summer I wrote about how much more interested I am in number theory than I used to be, and I think @_primes_ deserves a lot of the credit. Every hour, this bot tweets the next prime number. I enjoy scrolling through and retweeting the primes I find pleasing in some way.

In addition to helping me appreciate the aesthetic beauty of prime numbers, @primes has helped me understand the nine-less harmonic series. When I last checked, of the 10 most recent prime numbers it had tweeted, 6 of them had a 9 in them. If you select any digit and scroll through the numbers, you’ll find the digit more often than your intuition might lead you to believe. You might have runs where one digit doesn’t appear for a while, but those are balanced by the long strings where they do appear. (For example, all the recent primes have a 2, 5, and 6 in them, and they will stay that way for a while.)

After noticing how many times each digit pops up on the prime number tweetbot, the idea that most big numbers do contain any individual digit just clicked into place. I'm actually not sure how I feel about that. On the one hand, it's satisfying to feel like I finally understand something that used to mystify me. On the other hand, there's one less thing in the world that makes me say "What the balls?!"