Life of Prime

I was bored and decided to investigate prime numbers for half an hour, as you do.

I downloaded the first 1000 primes from the internet somewhere and graphed them against their position in the list. I expected an upward curve roughly resembling an exponential, which is what I got.

There seems to be some periodicity (loose) between large and small gaps between prime numbers.

I then wondered about the distribution of the distance between prime numbers, this was interesting; by far the most common separation was 6.

This made me think about how things would look in base 6 as this may have an important role to play: it does. See http://en.wikipedia.org/wiki/Senary

In Base 6, Senary or Heximal, as I have found it called, all prime numbers finish with a 1 or 5. Specifically they are all part of a set defined as 6n+1, apart from the first few.

At first this sounded very interesting to me, perhaps there is a pattern, I thought; but on reflection it's kind of obvious that this would be the case. If we take the base 6 units. 0,1,2,3,4,5 and also 10 (6 becomes 10 in base 6) we can see why:

Anything ending in a zero is a multiple of 10.

Anything ending in a 2 or 4 is a multiple of 2.

Anything ending in a 3 is a multiple of 3 :¬)

In base 6; 3 is like 5 in normal decimal, so 5, 10, 15, 20 are all multiples are 5 despite every other one being an odd number, in base 6; 3, 10, 13, 20, 23 are all multiples of 3 as 10 is a multiple of 3.

So all that is left is those ending in 1 or 5 (in base 6).

Some more info here: http://www.dozenalsociety.org.uk/pdfs/primeforms.pdf

I did decide to read up on Base 6 and its applications. It seems to be a very sensible numbering system and makes quite a few things easy to do, similar to a clock face calculating. Base 12 is similarily useful.

There is an article here from a proponent of base 6 mathematics: http://www.shackite.com/base-six/base-six.htm