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!

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