# Powers, Logic, and Prime Numbers

Yesterday we had quite an adventure with Graham’s number.  I wanted to have one more talk about powers before the weekend was over,  so the topic I chose for today was Fermat’s Little Theorem.   I wasn’t planning on going into too much depth on the number theory side, but I did want to emphasize a little bit of logical thinking and just walk through a few basic computations with powers.

We started with a quick introduction to Fermat’s Little Theorem and then computed an easy example with the number 5.  After that we talked a little bit about the logic – prime numbers satisfy Fermat’s Little Theorem, so if a number fails to satisfy Fermat’s Little Theorem, then it must not be prime:

Next we moved on to a couple of slightly more complicated examples.  First we took a look at 21 and tried to find the remainder when $2^{20}$ is divided by 21.  I told them that we only needed to worry about the remainders, which is fine for today.  We’ll cover the details behind  that idea sometime later on when we study number theory.  It was fun, though, just to show that you could compute the remainder when $2^{20}$ is divided by 21 without even knowing what the exact value of $2^{20}$ is!

We also discussed a number that satisfies the conditions of Fermat’s Little Theorem that isn’t prime – 1,729.  We didn’t go into the calculations, but did tell a fun and famous story about 1,729.

Finally, we couldn’t spend the morning talking about Fermat’s Little Theorem without mentioning Fermat’s Last Theorem. We walk through a few examples of triples that satisfy the Pythagorean Theorem and talk through what Fermat’s Last Theorem says.  We conclude by mentioning a few famous problems in math that remain unsolved.

Between yesterday’s discussion of Graham’s number and this one on Fermat’s Little theorem, it was  definitely a fun weekend of math.