# Continuing our talk about primes

Yesterday we had a fun talk about prime numbers starting with a problem I saw on twitter:

A neat problem from 5 Triangles and Dave Radcliffe

Today I wanted to spend a little time with my younger son revisiting some of the ideas from yesterday. I started by asking him to find 10 consecutive integers that weren’t prime. He remembered most of the ideas from yesterday, but there were one or two things that caused a little difficulty. This is a challenging argument for kids to follow since you are talking about numbers without really knowing what those numbers are.

In the first part of today’s discussion my son ran into the number 10! + 1. I thought it would be fun to use this number as a starting point to talk about (i) a simple proof of why there are infinitely many primes, and the related topic (ii) how to find numbers that we know contain “new” primes. We used Mathematica to make help out in this part of the discussion.

Oh, and one point I didn’t explain at all but definitely should have. Mathematica’s function FactorInteger[n] returns a list of primes that divide n as well as the highest power of that prime that divides into n. So, when we do the example FactorInteger[4! + 1] = FactorInteger[25] = {5,2}, the output “{5,2}” is saying that 25 = $5^2$ not that 5 and 2 divide into 25.

So, a fun follow up to yesterday’s project. Hopefully today’s talk helped a few of the ideas from yesterday sink in for my younger son.