Tag number theory

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.


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!


A terrific list of problems from Matt Enlow

Saw a great tweet from Matt Enlow today:

I decided to try out the first problem with my younger son this afternoon and we ended up having a really nice discussion. The problem is:

Is 11 the largest prime number that has all the same digits?

There’s a lot of great math ideas hiding in the problem.

Here’s how we got started – I love hearing the progression of ideas that he has all the way to the end of the movie:

By the end of the last movie he came to the conclusion that if there was a prime number larger than 11 that had all of the same digits, the repeating digit would have to be 1. So, the next thing we did was explore the first couple of numbers made with repeating 1’s:

After realizing that even 11,111 was going to be a challenge to try to factor we end upstairs to play on Mathematica. We made some pretty quick progress! Also, seeing the non-prime results on Mathematica also helped him see some patters in the non-primes that he hadn’t notices before – score one for some computer math!

Finally, I showed him the starting list on the Integer Sequence Database that shows the first few primes of the type we were looking at:

This was a really fun project – thanks to Matt Enlow for sharing this great list of problems!

Sharing John Cook’s Fibonacci / Prime post with kids

Saw a neat post from John Cook about using a fun fact about the Fibonacci numbers to prove there are an infinite number of primes:

Infinite Primes via Fibonacci numbers by John Cook

Funny enough, we’ve played with the Fibonacci idea before thanks to Dave Radcliffe:

Dave Radcliffe’s Amazing Fibonacci GCD post

That project was way too long ago for the kids to remember, so today we started by just trying to understand what the Fibonacci identity means via a few examples:

Next we looked at the idea from Cook’s post that we need to understand to use the Fibonacci identity to prove that there are an infinite number of primes. The ideas are a little subtle, but I think the are accessible to kids with some short explanation:

We got hung up on one of the subtle points in the proof (that is pointed out in the first comment on Cook’s post). The idea is that we need to find a few extra prime numbers from the Fibonacci sequence since the 2nd Fibonacci number is 1. Again, this is a fairly subtle point, but I thought it was worth trying to work through it so that the boys understood the point.

Finally, we went upstairs to the computer to explore some of the results a bit more using Mathematica. Luckily Mathematica has both a Fibonacci[] function and a Prime[] function, so the computer exploration was fairly easy.

One thing that was nice here was that my older son was pretty focused on the idea that we might see different prime numbers in the Fibonacci list than we saw in the list of the first n primes. We saw quickly that his idea was, indeed, correct.

This project made me really happy 🙂 If you are willing to take the Fibonacci GCD property for granted, Cook’s blog post is a great way to introduce kids to some of the basic ideas you need in mathematical proofs.

An AMC12 algebra problem that gave my son trouble

The problem below gave my son some trouble this morning:


When he got home from school we talked about it in more detail and it seemed to make more sense for him than it did this morning. The problem is a nice introductory algebra / quadratic problem:

Next I showed him a similar solution, but where “x” represented a different number:

Finally – just for a completely different way of looking at the problem – I wanted to show him a way that we could use the choices to help us find the solution. This is sort of cheating, but he was very confused by the problem this morning and I wanted to show him a way to get a little un-stuck when you are stuck.

Also, we got interrupted by the guy servicing our furnace – so sorry the video jumps in the middle 🙂

Sharing Kelsey Houston-Edwards’s Pigeonhole Principle video with kids

The 3rd video in Kelsey Houston-Edwards’s amazing new series was published last week. I’ve already used the first two videos for projects with the boys – I love this series so much!

Sharing Kelsey Houston-Edward’s [higher dimensional spheres] video with kids

Sharing Kelsey Houston-Edward’s Philosophy of Math video with kids

the latest video is about the Pigeonhole Principle and begins with the question – Do any two human beings have exactly the same number of body hairs:

Before diving into the video I asked the boys what they thought about the hair question – fortunately I got two different answers!

Next we watched Houston-Edwards’s new video:

Here’s how the boys reacted to the video:

(1) They were excited about the hair result and were also able to understand and explain it.

(2) They gave a nice summary of the Pigeonhole Principle.

(3) They really liked the example about 5 points on a sphere, so we took a really close look at that example. One of the tricky parts of that problem is understanding *why* you can draw an equator through any two points – both kids gave nice explanations of that idea.

Now I moved on to a couple of fun Pigeonhole Principle examples that weren’t covered in the video. I wanted to show the boys that the idea comes up in lots of different situations, including some that are not at all obvious Pigeonhole Principle situations!

The first example comes from my college combinatorics textbook – Applied Combinatorics with Problem Solving by Jackson and Thoro:

Screen Shot 2016-12-03 at 8.47.57 AM.png

Small twitter math world fun fact – the professor for this class (~25 years ago!) was Jim Propp!

Here’s the problem (which is example 5 on page 35 of the book):

Suppose that we are given a set X of 10 positive integers, now of which is greater than 100. Show that there are two disjoint nonempty subsets of this set whose elements have the same sum.

I had to do a little bit of work on the fly to translate the problem into something that the boys could understand (and also explain quickly why there are 1024 subsets), but it seemed like they enjoyed this example:

The last problem is one I remembered when reading through some of the other examples in Jackson and Thoro’s book and is one that I talked about with the boys last year:

A challenging arithmetic / number theory problem

Here’s the problem:

Show that every positive integer has a multiple whose base 10 representation consists of only 1’s and 0’s.

It certainly isn’t obvious at all at the start why this is a Pigeonhole Principle problem!

As I said at the beginning – I love this new series from Kelsey Houston-Edwards. I’m so happy to be able to use these videos to explore fun mathematical ideas with my kids!

Struggling through a challenging AMC 10 problem

This problem gave my son some difficulty yesterday – it is problem #19 from the 2011 AMC 10a


Last night we talked through the problem. The talk took a while, but I was happy to have him slowly see the path to the solution. Here’s his initial look at the problem:

Next we looked at the equation y^2 - x^2 = 141. Solving this equation in integers is a nice lesson in factoring. Unfortunately by working a bit too quickly he goes down a wrong fork for a little bit.

In the last video we found that the original population of the town might have been 484, and it might have changed to 634 and then once more to 784. We had to check if 784 was a perfect square.

Finally, we needed to compute the approximate value (as a percent) of 300 / 484. The final step in this problem is a great exercise in estimating.

So, a really challenging problem, but also a great problem to learn from. We went through it one more time this morning just to make sure that some of the lessons had sunk in.

Revisiting James Tanton’s base 3/2 exercise

Several years ago we played around with James Tanton’s base 3/2 idea:

Fun with James Tanton’s base 1.5

A tweet from Tanton reminded me about his project earlier this week. I was excited to revisit it and got a double surprise when my older son told me that he actually did it in his 7th grade math class last week! It is nice – actually amazing – to see Tanton’s work showing up in my son’s math class!

An unfortunate common theme with some of our recent projects is that they aren’t going as well as I hoped they would. Still, though, this was fun and I’ll have to spend a bit more time thinking about the last bit – how to write 1/3 using base 3/2.

We started by reviewing base 2 and, in particular, how you can play around with binary using blocks.

Next we looked at base 3/2. I’m sorry that this video runs 10 min – I definitely should have broken it into 2 pieces.

Finally we accidentally walked into a black hole. I assumed that writing 1/3 in base 3/2 wouldn’t be that difficult and that an easy pattern would emerge quickly. Whoops.

Turns out that no pattern emerges quickly, and even playing around on Mathematica for a bit after we turned off the camera we couldn’t find the pattern. The discussion facilitated by the work on Mathematica was great – at least my kids learned that (i) there are multiple ways to write a number in base 3/2, and (ii) there are easy sounding project that I can’t figure out!

I hope to revisit this part after I understand it better myself. Any help in the comments would be appreciated.

I really like this project and am sad that a little bit of stumbling around by us might have obscured the beauty of Tanton’s idea. Hope we’ll be able to revisit it soon.

What a kid learning math looks like – a challenging base problem

My younger son has been working through Art of Problem Solving’s Introduction to Number Theory book this summer. The topic for the last few weeks has been arithmetic in different bases. Today he can across a problem that gave him a lot of trouble. He worked on it alone for about 15 minutes and then we talked about it.

Here’s the problem:

In a certain base (12)*(15)*(16) = 3146. What is that base?

Here’s the first part which summarizes his initial thoughts on the problem. The work he does here shows that the base used in this problem must be lower than 10. Once he discovers that fact we talked about a few other ways that we could have seen that the base wasn’t 10.

Next we tried to see how we could identify the base we were looking for using some of the ideas from the last video. We used the last digit idea to eliminate 7 and 8, but the last digit idea told us that base 9 might actually work.

Then we did the arithmetic to show that we were indeed looking for base 9.

So, a really challenging problem, but a fun talk for sure. Working through problems like this one are a great way to review arithmetic and a neat way for kids to learn some basic ideas in number theory.