Thoughts on prime numbers

Many years ago I went on a residential course for Maths teachers. Apart from the Maths, we took part in a range of imaginative activities: one I remember best was playing card games where the four suits were square numbers, cube numbers, triangle numbers and prime numbers. So the 4 of squares, for example, would look like the image below, because 16 = 4^2. The 7 of cubes would be 343 – you soon got to know cubes up to 13^3. Triangle numbers are easy to calculate: the nth triangle number can be calculated as  \frac{n(n+1)}{2}  giving the fourth triangle number, for example, as 10. But playing with the cards you have the much harder task of working backwards: which triangle number, for example, has value 36?

CardBut it was the primes which were the killer, because there is no pattern to them. The Jack of primes? It’s 31 (the 11th prime), but there’s no formula for it. You just have to count on your fingers, which we did, and even playing a simple game like Snap was torture! Is 23 prime the same as 55 triangle – try and work it out and see how long it takes. The answer is no, because 23 is the 9 of primes, 55 is the 10 of triangles! One late night group tried playing Bridge – it was hilarious!

Many books have been written about prime numbers. Why are they so important? Partly because, in the same way that the letters of the alphabet form the building blocks of words, so every number can be written uniquely in terms of prime factors. 60, for example, is 2^2\times 3\times 5. Prime factors enable us to do things such as calculating lowest common multiples and highest common factors, and so are the basis for building the theory of numbers.

No-one has yet found a quick way to split large numbers into their prime factors, a property which is at the heart of modern cryptography. Even the fastest computers would take years to get to an answer, hence producing an effectively uncrackable code.

PrimeChartIf you were to work out prime numbers in ascending order, you would expect the gaps between them to get ever larger since there are more possible prime factors below them the higher you go. But, although they do thin out, it’s not by very much. The bar chart shows how many primes there are in every group of 200 numbers up to 10,000, the orange line showing the running mean. And, surprisingly, you can easily find prime numbers just two apart wherever you look: for example, 200,927 and 200,929 are both prime.


Puzzle answer: in my last blog I set the logic puzzle of a man looking at a portrait, saying: “Brothers and sisters have I none, but that man’s father is my father’s son.” Who is in the portrait? Because the question has a symmetrical feel to it, it’s very easy to think it’s a portrait of the man. But replace “my father’s son” by “me” and we get “that man’s father is me.” The portrait is of his son!

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