Times Tables and Prime Numbers (3)

I don’t know about you, but writing all this stuff to help you with your times tables and, I hope, lots of other interesting mathematical ideas too, is really doing my head in. But it’s so exciting and here’s another idea worth looking at.

My favourite topic in mathematics has to be prime numbers – I just love them. Had you guessed.

The reason I love them so much, I think, is that they break all the rules. In mathematics we are always looking for patterns of one sort or another, but prime numbers don’t seem to have many rules, unlike the times tables.

In fact, they are like the numbers left over when all the rules have been made. They are the ones that the rules don’t apply to!

You can get a flavour of this by thinking about the Sieve of Eratosthenes. All the multiples of 2, 3, 5, 7 etc are knocked out of the sieve (except these numbers themselves, of course), so all the numbers with rules are gone and we are left with all the rest. It’s fascinating.

Well, you can imagine that one of the things mathematicians have tried to do is to produce a formula that will give you just prime numbers and no others, just like the formula:

                                                          

gives you the triangle numbers.

As you will have guessed by now, no such formula for prime numbers exists (or if it does, we have not found it yet), but that hasn’t stopped mathematicians trying to find one and in this task we are going to look at some of their efforts to see how good they were.

You will need to have done a bit of algebra to do this work, but it is not too difficult if you have.

Let’s begin with this formula: 

                                                 

Try putting 1, 2, 3 etc in this formula in turn and see if the answers are prime numbers.

Let’s try 5 as an example:  52 + 5 + 41 = 25 + 5 + 41 = 71 and, sure enough, this is a prime number.

In fact, for the first 80 numbers for n, this formula produces a prime number in all but 7 of them. Can you find the first one that fails?

 

Here is another formula:

                                                               

When n =1, the formula gives 31, which is prime.
When n = 2, the formula gives 37, which is prime.

Again, which is the first number for n that does not produce a prime number?

I hope you have lots of fun investigating these.