Today I was looking for a project with my son and flipped open to the chapter on quadratic reciprocity. It had a few introductory ideas that I thought would be fun to share with my younger son.

We first looked at Wilson’s Theorem:

After Wilson’s theorem, we moved on to talking about perfect squares mod a prime. After a fairly long discussion here my son noticed that half of the non-zero number mod a prime are perfect squares:

Finally, I asked him to make a mod 11 multiplication table and we talked through some of the patterns in the table – including that the non-zero numbers had multiplicative inverses:

It was a really fun discussion today. I know next to nothing about number theory, but I really would like to use Weissman’s book more to explore some advanced ideas with the boys.

I started the project today by asking my son to tell me some things he knew about primes. He gave the definition of a prime numbers, explained how we know that there are infinitely many primes, and talked about twin primes, though he apologized for not knowing how to prove that there were infinitely many twin primes:

Next I showed him the polynomial and we talked about this equation producing a lot of primes.

Now we went to the “prime time” explorable and my son talked about what he saw in the first two examples -> the Ulam spiral and the Sack spiral.

Finally we looked at the last two patterns -> the Klauber triangle and the Witch’s spiral.

These problems from yesterday’s math contest looked like they would make a fun project, so had the boys work through the first 6 this morning.

Here’s problem #1 – this problem lets kids get in some nice arithmetic practice:

Here’s problem #2 – the challenge here is to turn a repeating decimal into a fraction:

Here’s problem #3 – this is a “last digit” problem and provides a nice opportunity to review some introductory ideas in number theory. The boys were a bit rusty on this topic, but did manage to work through the problem to the end:

Problem #4 is a neat problem about sums, so some good arithmetic practice and also a nice opportunity to remember some basic ideas about sums:

Next up is the classic math contest problem about finding the number of zeros at the end of a large factorial. My older son knew how to solve this problem quickly, so I let my younger son puzzle through it. The ideas in this problem are really nice introductory ideas about prime numbers:

The last problem gave the boys some trouble. BUT, by happy coincidence I’m about to start covering partial fractions with my older son, so the timing for this problem was lucky. It was interesting to see the approach they took initially. When they were stuck I had the spend some time thinking about what was making the problem difficult for them.

It inspired me to do a project on prime numbers with the boys. So, I grabbed my copy of Martin Weissman’s An Illustrated Theory of Numbers and looked for a few ideas:

We began by talking about why there are an infinite number of primes:

Next we moved on to taking about arithmetic sequences of prime numbers. There are a lot of neat results about these sequences, though as far as I can tell, they have proofs way beyond what kids could grasp. So instead of trying to go through proofs, we just played around and tried to find some sequences.

I also asked the boys how we could write a computer program to find more and they had some nice ideas:

Next we played with the computer programs. Sorry that this video ran a bit long. As a challenge for kids – why couldn’t we find any 4 term sequences with a difference of 16?

Finally, we looked at Evelyn Lamb’s joke to see if we could understand it!

It is definitely fun to be able to share some elementary ideas in number theory with kids!

Flipping through it last night I ran into an easy to state theorem that seemed like something the boys would enjoy exploring:

Wilson’s Theorem -> If is a prime number, then

The proof is a bit advanced for the kids, but I thought it would still be fun to play around with the ideas. There is an absolutely delightful proof involving Fermat’s Little Theorem on the Wikipedia page about Wilson’s Theorem if you are familiar with polynomials and modular arithmetic:

Here’s how I introduced the theorem. We work through a few examples and the boys have some nice initial thoughts. One happy breakthrough that the boys made here is that they were able to see two ideas:

(i) For a prime , is never a multiple of , and

(ii) There were ways to pair the numbers in (at least in simple cases) to see that the product was

Next we tried to extend the pairing idea the boys had found in the first part of the project. Extending that idea was initially pretty difficult for my younger son, but by the end of this video we’d found how to do it for the 7! case:

Now we moved on to study 11! and 13! At this point the boys were beginning to be able to find the pairings fairly quickly.

To wrap up this morning, I asked the boys for ideas on why we always seemed to be able to pair the numbers in a way that made Wilson’s theorem work. They had some really nice ideas:

(i) For odd primes, we always have an even number of integers – so pairing is possible,

(ii) We always have a product 1 * (p-1) which gives us a -1.

Then we chatted a bit more about why you could always find the other pairs to produce products that were 1. The main focus of our conversation here was why it wouldn’t work for non-primes:

Definitely a fun project! There’s some great arithmetic practice for the boys and also a many opportunities to explore and experience fun introductory ideas about number theory.

We are heading out for a little vacation before school starts and I wanted a gentle topic for today’s project. When I woke up this morning the idea of introducing the boys to quadratic reciprocity jumped into my head. The Wikipedia page on the topic gave me a few ideas:

I started the project by showing them the chart on Wikipedia’s page showing the factorization of for integers 1, 2, 3, and etc . . .

What patterns, if any, would they see?

Next we moved to a second table from Wikipedia’s page – this table shows the squares mod p for primes going going from 3 to 47.

Again, what patterns, if any, do they notice?

Now I had them look for a special number -> for which primes p could we find a square congruent to -1 mod p?

Finally, we wrote short program in Mathematica to test the conjecture that we had in the last video. Â The conjecture was that primes congruent to 3 mod 4 would have no squares congruent to -1 mod p, and for primes congruent to 1 mod 4 would, -1 would Â always be a square.

Sorry for the less than stellar camera work in this video . . . .

Saw a neat puzzle posted by Edmund Harriss last night:

Quick puzzle: Take two sets of numbers so that the sum of any two in either (allowing repeats) is in the other. If 1 is in one of the sets must they share a number?

I thought it would be fun to try it out with the boys this afternoon.

I didn’t give them much direction after introducing the puzzle – just enough to make sure that my younger son understood the situation:

After the first 5 minutes they had the main idea needed to solve the puzzle. In this video they got to the solution and were able to explain why their solution worked:

Definitely a fun challenge problem to share with kids. You really just have to be sure that they understand the set up and they can go all the way from there.

My son is finishing up a chapter on exponentials and logs in the book he was working through this summer. The book had a big focus on e in this chapter, so I thought it would be fun to show him the proof that e is irrational.

I started by introducing the problem and then with a proof by contradiction example that he already knows -> the square root of 2 is irrational:

Now we started down the path of proving that e is irrational. Â We again assumed that it was rational and then looked to find a contradiction.

The general idea in the proof is to find an expression that is an integer if e is irrational, but can’t be an integer due to the definition of e.

In this part we find the expression that is forced to be an integer if e is irrational.

Now we looked at the same expression that we studied in the previous video and showed that it cannot be an integer.

I think my favorite part of this video is my son not remembering the formula for the sum of an infinite geometric series, but then saying that he thinks he can derive it.

This is a really challenging proof for a kid, I think, but I’m glad that my son was able to struggle through it. After we finished I showed him that some rational expressions approximating e did indeed satisfy the inequality that we derived in the proof.

Today I thought it would be fun to revisit the calculation of a few of the last digits of Graham’s number.

So, with no review, I asked my older son what he remembered about Graham’s number and then we talked about the surprising fact that you could calculate the last few digits even though you really couldn’t say much else about the number:

Next I asked my son about how he would approach calculating the last digit. He gravitated to the right idea -> modular arithmetic. The ideas were a little confusing to him, but I let him work mostly on his own.

We didn’t get to the end in this video, but you can see how the ideas start coming together.

In the last video he had made some progress on finding the last digit, but one piece of the argument kept giving him trouble.

BUT, he did have a correct argument – it just took him a minute to realize that he was on the right track.

Again, this is a nice example of how a kid works through some advanced mathematical ideas.

Next we went to the computer to begin looking at the last two digits of Graham’s number. The last two digits of powers of 3 repeat every 20 powers, so it was easier to use Mathematica to find the cycle than it was to do it by hand.

Here I just explain the short little computer program I wrote to him.

Finally, we tried to see if we could use the idea that the powers of 3 repeat their last two digits every 20 steps to see if we could find the last 2 digits of Graham’s number.

As we started down the path here, I didn’t know if we’d find those last two digits. But we did! It was a nice way to end the project.

Last night I got an interesting comment on twitter in response to my Younger son suggesting that we write the numbers in a circle – a suggestion that we didn’t pursue:

At the beginning, there was a suggestion to write the numbers in a circle, then draw lines between the pairs. It is worth making a prediction about how that will look. The result may surprise you.

So, today we revisited the problem and wrote the numbers in a circle:

Next I asked them to try to find another set of numbers that would lead us to be able to pair all of the numbers together with the sum of each pair being a square. The discussion here was fascinating and they eventually found

This problem definitely made for a fun weekend. Thanks to Michael Pershan for sharing the problem originally and to Rod Bogart for encouraging us to look at the problem again using my younger son’s idea.