Tag number theory

One more trip through Goldbach’s comet with the kids

We’ve now down a couple of projects on the latest Numberphile video on the Goldbach Conjecture:

Sharing Numberphile’s Goldbach Conjecture video with kids

Exploring the Goldbach Comet with kids

Following those projects I thought it would be neat to let the computer run and find the numbers that could be written as a sum of two primes in many different ways (specifically, in more ways than any number less than it). Looking at those results produced a nice surprise:

An unexpected surprise for me in the Goldbach Comet

A double surprise was that Numberphile had just (the day before) published a follow up Goldbach Conjecture video that talks a little bit about the idea that explains the pattern I was seeing:

Last night I walked the kids through some of the ideas. We first watched the end of the Numberphile video and then talked about it briefly.

Also, I was pretty under the weather yesterday, so sorry for the low energy from me in this project:

Next we moved on to looking at the Goldbach Comet and told them about the project I was looking at while they were up in New Hampshire hiking.

They noticed the same pattern that I saw and I showed them the prime factorizations of a few of the numbers on my list.

After we talked about the factoring, I wanted to show them another surprise – the Goldbach Comet looks surprisingly symmetric around the numbers that can be written as the sum of two primes in lots of ways.

Fianlly, we wrapped up the project by looking at the symmetry I mentioned above a bit more carefully. I’d like to explore this symmetry a bit more myself!

We’ve really had a fun set of projects on the Goldbach Conjecture. It is definitely accessible to kids and a great way to show them an unsolved problem in math!

Exploring the Goldbach Comet

My wife and kids are going hiking today and I was looking for any fairly light project to do with the boys before they left. This morning I thought playing around with the Goldbach Comet would be a fun idea. We learned about it last week in Numberphile’s Goldbach Conjecture video:

Our first project from that video is here:

Sharing Numberphile’s Goldbach Conjecture video with kids

Today’s project needs a little disclaimer . . . . Sometimes when I decide to try something at the last minute things actually work out ok. Today was much more stumbling around than usual, unfortunately. But we had fun exploring anyway.

So, we started with some simple Mathematica code to explore the number of ways to write an even integer as the sum of two primes:

I gave the boys a challenge of finding the largest even number that can be written as the sum of two primes in 6 different ways. Then we played around a bit more – stumbling around aimlessly . . . .

Finally we used a program from the Wolfram Demonstrations Project to play around with the Goldbach Comet. That project we used is here:

The Goldbach Coment on the Wolfram Demonstrations Project site

I mainly used the code here to ask the kids what they thought they were seeing.

So, a fun project despite the numerous stumbles. I’d actually never heard of the Goldbach Comet prior to the Numberphile video. It was neat to play with.

Sharing Numberphile’s Goldbach Conjecture video with kids

Numberphile released a really nice video about the Goldbach Conjecture today:

I thought it would make an excellent project with the boys even though some of the ideas involving logarithms might be over their head. So, we watched the movie and then talked about some of the ideas that caught their eye.

Next we moved on to the individual ideas. The first one was the chart that David Eisenbud made at the beginning of the video. Drawing and then filling in this chart is a nice little arithmetic activity for a kid in elementary school.

Next we talked about logarithms. I started with an idea I learned from Jordan Ellenberg’s book “How Not to be Wrong” – the “flogarithm”. That idea is to oversimplify the logarithm by defining it to be the number of digits in the number. That simple (and genius) idea really opens the door to kids thinking about logarithms.

With that short introduction I explained what the natural logarithm was and moved on to some of the properties of primes that Eisenbud mentioned in the video (after fumbling with the calculator on my phone for a minute . . . .).

(Also, I noticed watching the video just now that I forgot to divide by 2 at one point – sorry about that.)

Finally, we checked a specific example – how many ways were there to take two primes and add up to 50? This part is about as far away from the complexity of logarithms as you can get – just some nice arithmetic practice for kids.

To warp up I asked them if they knew any other unsolved problems about primes. My older son mentioned something about twin primes. I showed the boys a simple argument (fortunately quite similar to the one Eisenbud gave in the movie for why there are lots of ways two primes can add to be a given even number) for why there ought to be infinitely many twin primes.

I think that kids are going to be naturally curious about primes. The Goldbach conjecture is one of the few unsolved problems that kids can understand. It was fun to share this video with the boys tonight.

Sharing Grant Sanderson’s “Pi and Primes” video with kids part 2:

Grant Sanderson’s latest video explaining a connection between pi and prime numbers is absolutely fantastic:

This video is sort of at the edge of what kids can understand, but it was fun to explore a few of the ideas with them even if understanding 100% of the video was probably not realistic. Our project on the first 10 min of the video is here:

Sharing Grant Sanderson’s Pi and Primes video with kids part 1

Also, we did a project on a different approach to the problem Sanderson is studying previously:

A really neat problem that Gauss Solved

I intended to divide our study of Sanderson’s video into three 10 minute sections, but the second 20 minutes was so compelling that we just watched it all the way through. After watching the last 20 min a 2nd time this morning I asked the kids what they found interesting. The three topics that they brought up were:

(i) The \chi function,

(ii) The formula for \pi / 4, and

(iii) Factoring ideas in the Gaussian integers

Following the introduction, we talked about the three topics. The first was factoring in the Gaussian integers. We talked about this topic in yesterday’s project, too.

Next we talked about the \chi function. I had no idea how the discussion here was going to go, actually, but it turned out to be fantastic. The boys thought the function looked a lot like “remainder mod 4”. Why it does look like that and why it doesn’t look like that is a really neat conversation with kids.

Finally we talked through the formula that Sanderson explained for \pi / 4. It probably goes without saying that Sanderson’s explanation is better than what we did here, but it was nice to hear what the boys remembered from seeing Sanderson’s video twice.

I love having the opportunity to share advanced math with kids. I don’t really have any background in number theory and probably wouldn’t have tackled this project with out Sanderson’s video to show me the path forward. It really is amazing what resources are on line these days!

Sharing Grant Sanderson’s “Pi and Primes” video with kids. Part 1

[This one was written up pretty quickly because we had to get out the door for some weekend activities. Sorry for publishing the un-edited version]

Grant Sanderson has a new (and, as usual, incredible) video on “Pi hiding in prime regularities”:

By coincidence, we’ve done a project on this topic before:

A really neat problem that Gauss Solved

The old project is based on Chapter 8 from this book:

Ingenuity Pic

Sanderson’s new video is pretty deep and about 30 min long, so I’m going to break our project on his video into 3 pieces. Today we watched (roughly) the first 10 min of the video. Here’s what the boys took away from those 10 min:

The first topic we tackled today was how to write integers as the sum of two squares. This topic is the starting point in Sanderson’s video and the main point of the project from the Ingenuity in Mathematics project. We explored a few simple examples and, at the end, talked about why integers of the form 4n + 3 cannot be written as the sum of two squares:

Next we turned our attention to the complex numbers and how they came into play in (the first 10 min of) Sanderson’s video. My focus was on the Gaussian Integers. In this part of the project we talked about (i) why it makes sense to think of these as integers, and (ii) how we get some new prime numbers (and also lose a few) when we expand our definition of integers to include the Gaussian Integers:

To wrap up I mentioned the topic from the prior project. The question there is something like this -> since counting the exact number of ways an integer can be written as the sum of two squares is tricky, can we say anything about how to write an integer as the some of two squares?

Turns out you can, and that the average number of different ways to write a number as the sum of two squares is \pi. Pretty incredible.

[and, of course, I confused an n and n^2 in the video 😦 Looking at the prior project will hopefully give a better explanation than I did here . . . . ]

I’m always excited to go through Grant Sanderson’s video with the boys. He has an amazing ability to take advanced ideas and make them accessible to a wide audience. Sometimes making the topic accessible to kids requires a bit more work – but Sanderson’s videos are a great starting point.

Fawn Nguyen’s incredible Euclidean Algorithm project

Fawn Nguyen recently published an incredible blog post about a project related to the Euclidean Algorithm that she did with her students:

Fawn Nguyen’s “Euclid’s Algorithm

Fawn’s projects are usually very easy to do right out of the box, and this one is especially easy since you can just start with her pictures. So, we just dove in.

You’ll see from the comments my kids had that Fawn really has made using this blog post effortless:

Next I asked them to make their own shapes. They built the shapes off camera and then we talked about them.

At the end I asked them when they thought a shape would require 1x1x1 cubes.

After hearing their thoughts about relatively prime numbers at the end of the last video I asked them to make a shape that wouldn’t require 1x1x1 cubes to finish. Here’s what they made and why they thought it would work:

Such a fun project. Fawn’s work is so amazing. I love using her posts with my kids.

Sharing Kelsey Houston-Edwards’s Cryptography video with kids

I’m falling way behind on Kelsey Houston-Edwards’s video series, sadly. Her “How to Break Crytography” video is so freaking amazing that it needed to be first in line in my effort to catch up!

So, this morning I watched the video with the boys. We stopped the video a few times to either work through some of the math, or simply to just have me explain it a bit. Overall, though, I think this video is not just accessible to kids, but is something that they will find absolutely fascinating.

Here’s what my kids took away from it:

Next we went upstairs to write some Mathematica code to step through the process that Houston-Edwards described in her video. In this video we (slightly clumsily) step through the code and check a few small examples:

When I turned the camera off after the last video my younger son asked a really interesting question -> Why don’t we just use Mathematica’s “FactorInteger[]” function?

We talked about that for a bit in this video and then tried to use Shor’s algorithm to find the factors of a number that was the product of two 4 digit primes.

So, we had the camera off for a little over a minute after the last video, but the good news is that Mathematica did, indeed, finish the calculation. It was a nice (and somewhat accidental) example of how quickly this algorithm runs into trouble.

The cool thing, though, is that it did work 🙂

Definitely a fun project, though it does require a bit more computer power than most of our other projects. I’m happy to be catching up a little on Kelsey Houston-Edwards’s video series – it really is one of the best math-related things on the internet!

Sierpinski Numbers

I was trying (unsuccessfully) to track down a reference on the chaos game for Edmund Harriss and ran across an unsolved problem in math that I’d never heard of before -> the Sierpinski Numbers.

Turns out that Sierpinski proved in 1960 that there are infinitely many odd positive integers k for which the number:

k * 2^n + 1

is not prime for any positive integer n.

It turns out that the smallest known Sierpinski number is 78,557, though there are 4 smaller numbers for which no primes have been found, yet. Those numbers are 21181, 22699, 24737, 55459, and 67607.

There’s lots of info on the Sierpinski numbers on Wikipedia:

Wikipedia’s page on the Sierpinski numbers

Tonight I wanted to explain a bit about the Sierpinski numbers to the boys as a way to review modular arithmetic. I also thought it would be interesting to see how they thought you could attack a problem like this one – especially in the 1960s!

So, here’s how we got started – a bit of Sierpinski review and then an introduction to the theorem mentioned above. It isn’t the easiest thing for kids to understand, so I wanted to be extra sure they understood all of the parts:

Next we talked a bit about modular arithmetic and why it wasn’t too hard to see, for example, that lots of the number we were looking at were divisible by 3. The math work here is a great introductory modular arithmetic exercise for kids.

Next we went to Mathematica to explore the modular arithmetic a bit more. Once we had the idea with 3, it was a little easier to see why there were repeating patterns with the remainders mod 5. The fun part was that the boys were able to see that one out of every 4 numbers would be divisible by 5.

Finally, we looked at the problem a slightly different way and tried to see if it was easy or hard to see if 3 (or 5 or 7 or 9) was a Sierpinski number. Would we ever see primes?

This project was really fun – it is always neat to stumble on an unsolved problem that is accessible to kids. Also, I’d really love to know how Sierpinski’s proof went – sort of amazing that it took 8 years after the proof that there were infinitely many numbers with this property to find the first one!

A terrific prime number question from Matt Enlow

A great question from Matt Enlow inspired a super fun conversation with the boys last night:

Before diving in to the project, I’d really recommend thinking about the question – even just for a few seconds – just to see what your intuition tells you.

We started the project by looking at the tweet and trying to make sure that the boys understood the question. The question itself was harder for them to understand than I expected. One reason was that they weren’t used to thinking about ages in terms of days.

Next we went to Mathematica and wrote a little program using the “PrimePi” function which tells you the number of primes less than or equal to a number.

We played around a little bit. Their initial instinct was to zoom in on a specific number like 30 years old. There were some fun surprises since the number of primes between two numbers bounces around a bit. They also had some really interesting ideas about prime numbers.

Eventually they decided to check a range of ages.

At the end of the last video we decided to check a range of ages, and we did that with a “For” loop. Once we did that we found a couple of really fun surprises 🙂

Running the program over night, the largest age that I found was 179,676 years old! I doubt that’s the highest number, though, and I love that the boys thought that there might be infinitely many solutions to this problem.

Thanks to Matt Enlow for posing this problem!

Revisiting Stephen Wolfram’s MoMath talk

Last week Stephen Wolfram posted an incredible summary of his talk at the Museum of Math:

We did a project using some of the code here:

Sharing Stephen Wolfram’s MoMath talk with kids

I think the ideas from the talk can provide kids with a really wonderful opportunity to explore math. We’ll hopefully revisit the ideas many times!

Today’s exploration follows the same line of ideas that we followed in the first project. The procedure we are looking at goes like this:

(1) Start with the number 1, and proceed to step 2.

(2) Whatever number you get here, cycle the digits to the left -> so, 123 becomes 231, 1045 becomes 0451 (so just 451 for computations), 110110 becomes 101101, and etc . . .

(3) Now multiply the number from step 2 by a fixed number N and add 1.

(4) Take the output from (3) and return to step (2).

We look at the sequence of outputs from this procedure in base 2, 3, 4, and 5 today. Quite amazingly, Stephen Wolfram showed that this entire procedure could be done with some very short code in Mathematica. Here’s a pic of the short code and also patterns we see in the digits when we multiply by 1, 2, 3, 4, 5, 6, and 7 at each step when we reun the procedure above in base 4.

Wolfram.jpg

If this seems way too complicated I’m not explaining the procedure well enough – go back to our first post on the subject or to Wolfram’s blog. I promise you’ll see that the explorations are totally accessible to kids.

 We started our project today by revisiting the results in base 2 and looking for strange or unusual or really anything that caught our eye in the digit patterns.

Also, I’m sorry that the zoomed in shots are so fuzzy (so, the first minute here and basically all of the 4th video). I didn’t realize how bad the footage was until it was published. Even with the fuzziness, though, you can still hear how engaged this kids are and how interesting it was for them to explore all of the strange patterns:

For the 2nd part of the project we looked at the patters of the digits in base 3:

Then we looked at base 4 and immediately saw something that we’d not seen before:

So, having explored bases 2, 3, and 4 we went back to some of the patterns we’d seen and got a nice surprise – we were able to find structure in some of those patterns. This video is the exploration that led to us finding the pattern in base 2.

Again, I’m sorry this video is so fuzzy – wish I would have caught that when we were filming 😦

Now we moved on to exploring some of the patterns that we’d seen in base 3 and base 4 – that exploration allowed us to predict a pattern in base 5 even though we’d not yet looked at any of the digit patterns in base 5!

I can’t wait to play with Wolfram’s ideas a bit more. The ideas are such a great way to expose kids to exploration in math!