Prime Gaps and a James Tanton problem

Early this week James Tanton, a seemingly bottomless pit of great math ideas for kids, posted the following problem on Twitter:

A nice little twist on an old problem that inspired our Family Math talk for today.

The first thing that we did was do a quick review of prime numbers – what are primes, and can we list the first ten or so prime numbers?  After that we talked about the two types of “gaps” involving prime numbers.  One interesting, and still unsolved, question about prime numbers involves the number of “twin primes.”  So, are there infinitely many pairs of prime numbers like 3 and 5, or 11 and 13, that differ by 2.  An important step in answering this question was made last year by Tom Zhang of the University of New Hampshire who discovered (incredibly) that there are infinitely many prime numbers that are less that 70,000,000 apart from each other.   Not quite a difference of 2, but still amazing!

Almost the opposite question is the one posed by James Tanton – how large can the gap between consecutive prime numbers be?   That is the question that we’ll focus on for the rest of this talk:

Before moving on to answer the main question, though, in the last video my younger son mentioned that there are infinitely many prime numbers.   I thought it would be fun to show why that statement is true, so the next video walks through a simple proof that kids can understand.  I think (but have not verified) that this proof is attributed to Euclid.  In the course of this proof I also mention one reason why mathematicians do not like to consider the number 1 to be a prime number.

Finally we get around to discussing Tanton’s question.  We start by finding 1,000,000 consecutive non-prime integers and then move on to finding 1,000,000 consecutive odd prime numbers.  It was nice to see that my younger son was able to understand how to make the leap from all integers to just odd integers.

I think that there are lots of neat examples from number theory examples that kids will really enjoy.  The problem posed by James Tanton that we focused on today is a really fun problem to work through with kids.