Tag: number theory

Sharing Tom Apostol’s irrationality of the square root of 2 with my younger son

I saw a really neat tweet from Lior Patcher last week:



I thought it would be fun to share this proof with my younger son since the geometric ideas in it are both surprising and super interesting. Unfortunately this one didn’t go nearly as well as I’d hoped. I missed a good idea that he had and got caught up in a few details that weren’t that important. Oh well, even after 10 years of doing these projects, I don’t have a good feel ahead of time for how they’ll go.

That said, here’s what we did. I started by having him walk through what is probably the most common proof that the square root of 2 is irrational:

Next we looked at Apostol’s proof and talked about some of the geometric ideas, and I just 100% missed that he was absolutely on the right track:

Now we took a look at an algebraic approach to the problem using the Pythagorean theorem. This part also didn’t go as well as I hoped and I might revisit it tomorrow just to make sure that these algebraic ideas made sense:



Finally, we came back to the geometric ideas since I realized that he was on the right track. Unfortunately I spent way too much time at the end of this part on a minor point. But hopefully the main geometric idea that we talk through in the first half of this video came through ok.

It is always disappointing when these projects don’t go quite as planned – I definitely want to push the “try again” button on this one.

Sharing Andrés E. Caicedo’s amazing twitter thread on even / odd irrationality proofs with my younger son

Yesterday I saw an amazing twitter thread by Andrés E. Caicedo:

I thought that some of the ideas would be great to share with my younger son and started by asking him if he remembered the usual proof that \sqrt{2} is irrational:



Following the twitter thread, I asked him how he thought the proof that \sqrt{5} is irrational would go. He gave the proof that I think most math people would give:

Next we walked through the “new to me” proof in Caicedo’s twitter thread. The ideas are definitely accessible to kids. In addition to being accessible, the ideas also provide a nice way for kids to get some algebra practice while exploring a new math idea:



Finally, we talked about the surprise that this method of proof doesn’t work for \sqrt{17}. My son had an interesting reaction – since this method of proof doesn’t seem to rely on the underlying number, he was surprised that it didn’t work as well as the method he’d used for \sqrt{5}



I really loved talking through Caicedo’s thread with my son and am really thankful that he took the time to share this fascinating bit of math on Twitter yesterday!

Talking about some introductory number theory ideas with the boys

My younger son is working his way through Martin Weissman’s An Illustrated Theory of Numbers right now:

He’s in the chapter on greatest common divisor and least common multiple now, and I thought talking through some of the ideas he’s seeing would make for a good project this morning. It gave him a chance to talk about what he’s learning and it gave my older son a chance to review some ideas he’s seen before.

We started by talking about the Euclidean Algorithm:

Next we discussed the interesting identity that the product of the LCM and GCD of two positive integers is equal to the product of those integers:

Now we moved on to discussing how the ideas we talked through in the prior videos could help us solve Diophantine equations. Here my younger son introduces the main ideas:

To finish, I had my older son explain why the general solution my younger son introduced in the first video was



I can’t say enough good things about Martin Weissman’s book – it has really gotten my son interested in number theory. Can’t wait to explore more of the ideas in the book with him!



Having kids play with 4 and 5 dimensonal Hasse diagrams

Yesterday we did a fun project on Hasse diagrams:

https://mikesmathpage.wordpress.com/2020/05/24/exploring-hasse-diagrams-with-kids-thanks-to-martin-weissmans-an-illustrated-theory-of-numbers/

The boys got the hang of a few relatively simple examples but also noticed that going to numbers with 4 prime factors would get pretty hard to draw.

After we finished the project I saw a post on twitter about a 5d cube and was reminded that we had a 2d projection of a 5d cube hanging on our living room wall:

So for a challenge project this morning I had the boys try to figure out how a Hasse diagram would work in 4 dimensions and in 5 dimensions.

Here’s how the 4d case went:

The 5d case was significantly more challenging – but they got there! Here’s the explanation of their work:



Who would have ever thought that a 5d cube appearing in your twitter feed would be exactly the thing you needed to see for a new math project!

Exploring Hasse diagrams with kids thanks to Martin Weissman’s An Illustrated Theory of Numbers

I’ve just started the book An Illustrated Theory of Numbers by Martin Weissman with my son:

We are going slowly and are just a few pages in, but I wanted to so a project with Hasse diagrams today because he told me last week that seeing those diagrams in the beginning of the book is what made him want to study the book a bit more.

We started today by looking at the book and exploring a bit about factoring integers:

After that introduction I had the boys read the section on the book on Hasse diagrams (roughly 1 page long) to be sure they understood how they worked. Here’s what they had to say and then a bit of practice:

It turned out that the final exercise in the last video – writing the Hasse diagram for 36 – proved to be a little tricky for my younger son. Because the last video was running long we broke things into two parts. Here we finish the diagram for 36:

We finished up by looking at one of the Hasse diagram exercises in the book. Here the boys wrote the diagrams for 7, 15, 18, and 105.

This project was a nice light touch one. It gave the boys an opportunity to review a bit of arithmetic and introductory number theory. It was also fun to explore this interesting connection between number theory and geometry.

Finding integers that can be written as the sum of two positive squares in exactly 7 different ways

A few days ago we did a project using Mathologer’s amazing video on Fermat’s “two squares” theorem. At the end of the project the boys were wondering about why so many of he numbers we found that could be written as the sum of two squares in several different ways were multiples of 5. I was wondering the same thing and spent two days playing around and trying to learn more these sorts of numbers. Even after searching the positive integers up to 3,000,000, all of the numbers I found that could be written as the sum of two positive squares in exactly 7 ways were multiples of 5. What was so special about 5?

Overnight I got some great twitter advice on the subject form Stephen Morris and Alex Kontorovich. Their ideas helped me understand a bit more about what was going on. Tonight I explored some of the basic ideas with the boys. I know next to nothing about the number theory here, but am completly amazed by the never ending patterns that are hiding inside of the integers!

We started today’s project by looking at all of the positive integers less than 1,000,000 that can be written as the sum of 2 positive squares in exactly 7 ways. Here’s what they noticed:

At the end of the last video my younger son thought that it might be useful to factor all of the numbers on our list. We did that off camera and then the boys looked for patterns in the numbers and factors. Finding patterns in the factored numbers was more challenging than I expected, but they were able to make some progress.

Based on what we noticed we took some guesses at numbers that were not multiples of 5 that could be written as a sum of two positive squares in exactly 7 ways.

Finally, we used the Wolfram Alpha code that Stephen Morris showed us to check if the numbers we guessed really could be written as the sum of two positive squares in exactly 7 ways.

This project was incredibly fun. It shows how computers (and Twitter!) can really help kids explore some pretty advanced ideas. I’m really interested to see how we might be able to explore a few more related ideas in the next week.

Using Mathologer’s video about Fermat’s Two Squares theorem with kids

Yesterday we watched this fantastic video by Mathologer on Fermat’s Two Squares theorem:

I’m hoping that we can do a couple of different projects based on the video. Today we talked about some ideas from the video and wrote a project to explore sums of squares.

We began by discussing some of the ideas in the video that the boys found interesting, and then talked through a few of the proof ideas from the vdeo:

Next we checked out one of the algebraic identities that came up in the Mathologer video. I thought that checking these identities would be good algebra practice.

Finally, we went to Mathematica to write a program to look at sums of squares. There was one slightly tricky part of the program that took a minute or two for the boys to explain. I thought that they had understood this concept while we were writing the program, so it turned out to be sort of lucky that we went back to talk about it.

Anyway, here’s the program that we wrote and a little bit of play with the numbers. The surprise was that it seemed like almost all of the numbers that could be written as sums of squares in lots of ways were multiples of 5. We might explore this idea a bit more in a later project.

Fermat’s Two Squares is really fun to explore with kids – and Mathologer’s video is a great way to show kids an proof that they can understand. I’m excited to explore the ideas a bit more with the boys later this week.

Having the boys learn from Mathologer’s Fermat’s Christmas Theorem video

Mathologer put out a fantastic video last week:

I had the boys watch the full video and come up with two things that they thought were interesting. Here they explained their choices and gave a few thoughts about the video:

My younger thought the approximation of a circle by 1×1 boxes was interesting. Here we talked about that idea and sort of hand waved why the approximation gets good:

My older son thought that the concepts of the “good” and “bad” numbers was interesting. I let my younger son do a lot of the talking when we were talking about sums of squares. It was also interesting talk about why the proof that none of the integers of the form 4n + 3 can be written as the sum of two squares was easy, but the proof that all integers of the form 4n + 1 can be is hard.

I hope to return to the more complex questions the boys found interesting in a different project. Maybe next week!

A neat unsolved problem in number theory that kids can explore

Yesterday I saw a really neat thread on the Collatz conjecture from Alex Kontorovich

In that thread is a blog post by Alex’s friend Igor Park and Park’s blog post as a link to a neat set of lecture notes by Barry Mazur. AND, in Mazur’s notes is this “new to me” unsolved problem in number theory:

Mazur

Instead of continuing on our journey through Mosteller’s 50 Challenging Problems in Probability, I decided to explore this problem with the boys today.

Here’s the introduction to the problem and a bit of playing around with a few of the small cases:

In the last video the boys thought that the squares would all have to be odd and the primes would have to be odd. Here we explored both of those conjectures. That exploration led to a discussion of why odd numbers always have squares that are congruent to 1 mod 8:

Now we continued the discussion from last video and investigated the primes that could appear in this problem. We started by showing that 2 could never appear and then eventually found that only primes of the form 4k + 1 could appear:

Next we moved to the computer to explore more cases of the conjecture. This was mainly an exercise into writing a simple program in Mathematica, but it led to an interesting discussion as well as an idea for further exploration:

Finally, we modified our program to explore the number of different solutions to the problem for each number. The modification to the program was actually really easy and the histogram was fascinating to see:

It is really fun to be able to explore an unsolved problem with kids. I especially love unsolved problems that allow kids to get in some secret arithmetic practice will getting a bit of exposure to some advanced ideas in math. Seeing this problem yesterday and getting to explore it today with the boys was a real treat!

A neat property of reciprocals of primes

I’d pulled out Ingenuity in Mathematics by Ross Honsberger yesterday in a twitter thread about old but fun math books. It was still on my dining room table this morning when I was looking for a project.

Chapter 16 showed a neat idea that I’d never seen before – if the decimal expansion of the reciprocal of a prime number has a repeating pattern with an even number of digits, then the first half of the digits plus the last half will add up to a number with all 9’s.

An example with 1/7 shows the property:

1/7 = 0.142857142857142857…., and

142 + 857 = 999.

The proof of this fun fact was a little more than I wanted to get into today, so instead I talked about reciprocals, then showed the property, and finally talked about Fermat’s Little Theorem which is one of the key elements in the proof of this property of prime reciprocals.

Here’s how we got going – just an introductory talk about repeating decimals:

Next up was the repeating decimal property of some prime numbers. It was neat to hear what the boys thought about this property:

Finally, we talked a little modular arithmetic and about Fermat’s Little Theorem.

This was definitely a fun and light project. I think the full proof of this interesting property of prime reciprocals is accessible to kids, but would take some planning. It was too much for today, though, but I was still happy with the discussion the property inspired.