Earlier in the week Robert J. Lemke Oliver and Kannan Soundararajan of Stanford announced a totally surprising discovery about prime numbers. Evelyn Lamb has a fantastic article about the paper here:
Peculiar Pattern Found in “Random” Prime Numbers
The new paper suggests that the prime numbers are not distributed quite as randomly as mathematicians had previously expected. In particular, the last digit of consecutive prime numbers has a distribution that is different from what you’d expect if the distribution of primes was random.
I thought this would be a fun result to discuss with kids. One surprising thing about this result is that it is pretty easy to understand. In fact, you can double check the result on a computer really quickly.
Before jumping in to the result about primes, though, I wanted to spend a few minutes talking about probability and probability distributions. This part of the project turned out to have some extra fun when my older son asked a neat question about dice.
Here’s the introduction to today’s project and my son’s question – what is the probability of seeing at least one 5 when you roll two dice?
With the complimentary counting problem behind us, we moved on to talk a little bit about the difference between probability and probability distributions. Once again we had a little detour following a statement that my older son made. I was happy to have these extra little conversations about probability this morning:
The next part of the project involved talking about prime numbers. We talked a little bit about how mathemticians viewed the primes. Number theory and prime numbers are not my field – hopefully the details in this part are right. A great (and accessible) read about prime numbers and randomness can be found in Jordan Ellenberg’s How Not to be Wrong.
It was really fun to hear what the kids had to say about the last digit of consecutive prime numbers here. This problem is a great way to get kids thinking about math.
Now we moved to the computer and used a little Mathematica program to study what the distribution of the last digit of consecutive prime numbers looked like. We chose to look at prime numbers that ended in 1. It was neat to see the results. Loved hearing what my younger son observed: “it seems like there’s a lot less primes ending in 1”:
We wrapped up by looping through the first 25,000,000 primes and compared our results to the results that were given in Evelyn Lamb’s write up from above. Our results were really close – yay!
So, I think that talking about this new discovery makes for a really fun project for kids. I’m sure that our project could be improved quite a bit by any mathematician who had a good understanding of number theory (since my understanding of that subject is essentially zero!), but even the high level walk through that we did today was fun. It is pretty amazing to find a new discovery about prime numbers that can be understood by kids!