What I’ve learned playing around with primes

I’ve been playing around with the new result about primes by the Stanford mathematicians Robert J Lemke Oliver and Kannan Soundararajan. The result has been interesting to share with my kids:

Looking at the new discovery about primes with kids

Prime triples and the Sieve of Eratosthenes

and also to study on my own:

My fun interaction with prime numbers this week

I’ve kept going – slowly – after writing the last post. As I write this one I’ve counted ( via a simple Mathematica program) how often the last digits of consecutive prime triples occur up to the 14 billionth prime.

All of the data is in this google doc:

My prime triple counting data

On the “Prediction vs Actual” tab you can see the predicted results from the paper vs the actual counts for the last digits of consecutive primes (in base 10). I was able to understand the paper’s predictions for triples of the form (a,a,a) and (a,b,a).

For example (and assuming I’ve done the calculations right), in cell L9 of the Prediction vs. Actual tab, you’ll see that the new paper predicts 8,664,330 consecutive primes ending in (7,7,7) from the 7 billionth prime to the 8 billionth prime. There are actually 8,699,947 (cell L28). Not bad!

I’m interested to keep going to study the errors a bit more.

Here’s what I’ve learned so far:

(1) The new paper is clear enough for me to understand a few pieces even though I have virtually no background in number theory.

(2) Understanding those pieces took some work, though, and I’ve probably had to spend 5 hours to get the few pieces that I have gotten. I’m glad I didn’t give up 🙂

(3) In base 10, triples of consecutive primes end in (a,b,c) and (-c,-b,-a) with roughly equal frequency. I saw this result in the data and talked about it with the boys in the Sieve of Eratosthenes project above (after writing to the prime paper’s authors to ask for help understanding it).

(4) I have lots of ideas about how to understand all of this data about primes and essentially all of those ideas turn out to be totally wrong – ha ha.

(5) Working through the paper to calculate some of the paper’s predicted values for prime triples was one of the most satisfying activities that I’ve done in math this year . . . . even though I’m not sure that my calculations are right.

(6) One of the things that’s puzzling me now is that the actual vs predicted errors for the triples with equal digits – (1,1,1), (3,3,3), (7,7,7), and (9,9,9) – are too large. They seem to have size roughly proportional to 1/Log(n). The large errors make me think I’ve missed something in the paper – that’s the next thing I want to study.

(7) Working through any of the basic ideas about prime numbers from this paper is a great computer exercise for kids – especially if you have an easy way to identify primes (I was lucky to have Mathematica).

(8) I was really nervous about writing to the authors of the paper to ask about why the triples (a,b,c) and (-c,-b,-a) appear with equal frequency, but I’m glad I did. It was so cool that they wrote back.

So, this has been a really fun little side project for me. Many of the public results in math over the last few years have been great, but way way way over my head – the prime gap result from Zhang, the Fields Medal results, and the Breakthrough Prize results for example. I don’t know how frequently a new result in math or science can be understood by the public, but I sure am glad that this one landed on my desk!