One of the “nice” things about number theory is that it gives rise to some problems that are extremely easy to state and understand, but very hard to prove. Your children won’t be able to prove any of these, but they might have fun looking for counter-examples, and they might think it’s fun to work on something that professional mathematicians can’t solve, and that would definitely make international news if they found a counterexample.

### The Collatz conjecture

The Collatz function of a number x is given by dividing x by 2 if it is even, and multiplying it by 3 and then adding 1 if it is odd.

So, for example,

1 is odd, so C(1) = 3*1+1 = 4.

2 is even, so C(2) = 2/2 =1.

3 is odd, so C(3) = 3*3+1 = 10.

4 is even, so C(4) = 4/2 = 2.

We can iterate this function by applying it repeatedly – so, for example, 3->10->5->16->8->4->2->1->4->2->1->4->2->1…

Notice that in this case we ended up cycling round the sequence 4,2,1,4,2,1 etc for ever.

The collatz conjecture states that this will happen whichever number you start with. No-one knows if it’s true or not.

Pick some numbers to start with and see what paths they take.

Draw a graph of which numbers go where under this function – that is, a picture with arrows from 3 to 10, 10 to 5, and so on. Work out how to lay out numbers to get a neat graph.

Can you find a number less than 30 that takes more than 100 steps before it gets to the 4,2,1 cycle?

### The twin primes conjecture

If you don’t know what a prime number is, try this investigation first.

The twin primes conjecture states that there are infinitely many pairs of primes that differ by two – for example, 5 and 7, 11 and 13, 17 and 19, 29 and 31, and so on.

How many pairs of twin primes can you find? What do you notice about the numbers between them? Why is that?

### The Goldbach conjecture

Again, you’ll need to know what a prime number if for this one.

The Goldbach conjecture states that every even number apart from 2 can be expressed as the sum of two prime numbers.

Write down the prime numbers up to, say, 100.

For each even number up to 100, find a pair of prime numbers that sum to it.

For each even number up to, say, 40, write down /all/ the pairs of prime numbers that sum to it. Doing this by trial and error would be an awful lot of work, but can you think of a more efficient way to do it (hint: think about finding pairs of primes that sum to even numbers by looking at prime numbers rather than by looking at even numbers).

Can you find an even number that cannot be expressed as the sum of two prime numbers? (Warning: we know that any such number must have at least 19 digits, so looking at small numbers won’t work)