Pick any number. We can make that number with four square numbers.

We've been carrying on our work investigating numbers. As we mentioned, Pierre Fermat had said that any number can be made by adding four square numbers.

We picked numbers and tried it out.

 We've made a big display of this in EC1:

Can you fill any of the gaps?

We'll give you any number with just four squares

Pierre Fermat, who lived in Toulouse, wrote in 1638 that any number can be made by any one of these ways:


1. Add three triangle numbers.

These are the triangle numbers:

2. Add four square numbers.

These are the square numbers

49   36   25   16   9   4   1   0

3. Add five pentagonal numbers.

 These are the pentagonal numbers:

 4. Add six hexagonal numbers.

These are the hexagonal numbers:

We had a go at the easiest one: making lots of numbers with four square numbers. We all picked a number, and then thought hard about which combination of squares would make it (Zero is a square number too.)

Pierre Fermat, Holly, prime numbers and square numbers

Back in January Holly introduced us to an interesting fact about prime numbers and square numbers. This is something that was discovered by a famous mathematician from Toulouse called Pierre Fermat. The green prime numbers circled here can all be made by adding two square numbers together:

Today in 4G we checked this:

And here are all the ones we did:

One little correction...

More on Holly's post

Holly posted a fantastic post at the weekend about square numbers and prime numbers. If you haven't had a look at it yet, it's really interesting: have a look! Also read the comments.

It's certainly made me, Mr Gregg, think. In 4G today we looked at the numbers Holly's dad had talked about, and used it for our counting circle:

These are all the "four times table plus one" numbers, they are the " 4 times something + 1" that Holly's dad talked to her about.

We looked for prime numbers among these. We used a 100 square, but it would have been better on a hundred-rectangle like this:

All the prime numbers are circled. The ones that we're looking at are the green ones. (It never works with the red primes.)

Then we found that all of them could be made by adding two square numbers.

1    4    9    16    25    36    49     64    81

Can you do it? If you can make any of those green primes by adding two square numbers, write it in the comments below.