Sexy numbers

After a three month hiatus the 1729 blog is back! My last post was a little musing on my research project (Prime numbers) and so, six exams, three pieces of coursework and my research project later I have the time to think and write again. So welcome back to readers old and new and I hope you have learned my lesson – never start a new blog before assessment season!

We are living through an incredibly sexy time. Maths says so, so it must be true. Before I explain why, which numbers have the IT factor to be called sexy? 5 and 11 have it so do 7 and 13 and so does 17 and 23. Can you spot the pattern yet? These numbers are pairs of prime numbers* separated by gap of six numbers, and, as any school boy who was taught Latin knows, the Latin for six is sex hence, Sexy prime.

The interesting thing about sexy prime pairs is that when added together they are all multiples of four and we can prove it. As two is not prime (and no other prime number is even) we can say that the first prime number is 2n+1**

The second number will be 6 more than this so this will be 2n+1(+6)

So, adding these two together 2n+1+(2n+1+6) = 4n+8

We can factorise this to be 4(n+2) therefore the sum of each pair of sexy primes will be a multiple of four.....well, I find that interesting!

So, why are we living in an incredibly sexy time? 2011 and is a prime number and so is 2017. So, we all have the luck to be living right smack in the middle of a sexy pair of prime numbers. Can you feel the sexiness? lucky us!

When I started my project on primes a friend of mine commented that it was a fairly pointless project with no real world application. She said that as if it is a bad thing! In fact one of Britain’s greatest mathematicians G.H Hardy would have revelled in this comment. In his book ‘mathematicians apology’ he wrote  “No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.”

For much of history prime numbers were seen as a mathematical curiosity with little or no practical use. Du Sautoy (2003:48) notes an instance of where “Tables of primes were considered so useless...that they ended up being used for cartridges in Austria’s war with Turkey.”***

As the twentieth century progressed however practical uses for prime numbers became apparent including, for example, uses in cryptography and most secure systems from e commerce to secret service codes rely upon the use of large primes. This transformed the search for primes from a purely academic curiosity to one of great interest for big business and government with many organisations (such as the electronic frontier foundation) offering large rewards for finding large prime numbers over one hundred million digits long.

In short, for hundreds of years prime numbers were neither use nor ornament ‘in the real world’ and then it all changed in a matter of a few years. Now, the world turns on prime numbers. I like this and it is a great answer to the people who question why we spend money on theoretical research and activities with no apparent benefit ‘in the real world.’ So let the mathematicians, scientists and explorers do their thing. Let them explore the world and see where their curiosity takes them for unless you undertake the journey you never know what you may find

* A prime number has only two factors: 1 and the number itself

** If an even number is divisible by 2 then it stands to reason that an odd number is a multiple of 2 +/- 1.

*** Du Sautoy 2003. The music of the primes. Harper Collins.

Update********

A friend of mine has just asked 

Isn't it true that the total of any two odd numbers 6 apart will be divisible by 4, primes or not? 

Quite true. It's actually a better description of the proof I used above