I'm not blaspheming here. This really is amazing. It's a lovely example of mathematics. I read about it today. I never knew it before:
The prime numbers below 100 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
I have highlighted some of them. I'll tell you why a bit further on. You might like to try and work it out. It would be hard to guess.
If you divide up the highlighted ones from the others you get two lists of numbers
Bold prime numbers: 2, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97
Shy prime numbers: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83
Do you notice anything about them? The bookb hinted: "Try subtracting each number in the list from it's neighbour". I still didn't get it. I tried subtracting numbers in the different rows. Duh!
When you subtract the numbers for the first row you get
Bold: 3, 8, 4, 12, 8, 4, 12, 8, 12, 16, 4
and for the second row you get
Shy: 4, 4, 8, 4, 8, 12, 4, 12, 8, 4, 8, 4
I'm sure you've spotted it now. Apart from the naughty 3 at the beginning of the first row, all the numbers are multiples of 4: 1x4, 2x4, 3x4 etc.
And that rule applies to every other number in the sequences that you care to try.
And mathematicians have proved that:
1) Both sequences (both bold and shy prime numbers) go on forever
2) The further you go the closer the ratio of bold to shy gets to 1:1
3) All the numbers (apart from that naughty 3) are multiples of 4. (In fact they are more precise than that)
Oh, I haven’t told you how the prime numbers were divided into bold and shy ones. Simple: Bold ones are sums of two squaresc
2=12+12, 5 = 12+22, 13=22+32, 17=12+42 ...
the others are not.
So why is it that there is a relationship between multiples of four, prime numbers and sums of squares of numbers?
That old God chappy was pretty devious when he was creating the laws of mathematics and he sneaked that one in.
Or did God create the laws of mathematics? Or were they always out there? Please, one of my religious brethren, ask your priest and let us know.
Foot notes:
Just in case you have forgotten / didn't know
a) Prime numbers are whole numbers that cannot be divided by other whole numbers other than 1 and themselves.
b) The book is "I am a Strange Loop" by Douglas Hofstadter. Lent to me by David.
c) A square, in this context, is one whole number multiplied by itself. E.g.
two squared is written 22=2x2=4,
five squared is written 52=5x5=25.
The sum of two squares is what you get when you add two squares together.
I continue to be fascinated by things I don't understandPosted by: Mum | November 23, 2010 at 05:44 PM
Interesting but is it entirely surprising?All primes are odd (apart from 3) therefore the difference between any two primes will be a multiple of two. So of course you can split them into two groups both of which have two between them. I will bet you £100 that you can split them into 4 groups where the difference is between the numbers in each group is 8. I'd also bet (£50) that because primes are as good as random then as the series approaches infinity then the number of numbers in each group tends to 1:1:1:1.So to the last bit. Why can 4x+1 be described as a sum of squares, is this always the case or is it just for all the 4x+1 numbers that are primes. And conversely why is all of this not true for 4x+3 numbers?Posted by: Trevor the Younger | November 23, 2010 at 03:19 PM
There was an old chappy from heaven,Who asked "What is 30x7?"If it isn't a prime,You can give me a dime, And we'll send in our 2nd 11.Posted by: Mum | November 20, 2010 at 10:31 PM
No commentPosted by: Mum | November 20, 2010 at 03:34 PM
Reading this at the end of a VERY long week and I have to admit I'm not up to it. There's life in there somewhere, but not as I know it!!I'll have to come back to this when less tired.Posted by: Paul | November 19, 2010 at 06:45 PM

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