## Re: [PrimeNumbers] Can someone help explain this pattern

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• ... Phil probably thought you were plotting numbers in a spiral but that is not the case. You are plotting them like this (possibly mirrored/rotated and I
Message 1 of 3 , Sep 22, 2008
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Phil Carmody wrote:
> --- On Mon, 9/22/08, Gav C <g4v83@...> wrote:
>> For background, I wrote a small java program that plots
>> prime numbers on a grid. I tried a few different grid sizes.
>> First I tried 100 by 100, I found lots of straight lines -
>> but realized that these corresponded to the even numbers and
>> multiples of 5.
>>
>> Now I've got a grid that's 546 across, and I find 2
>> straight lines that are 9 numbers wide, at these widths 241
>> to 249 and then 295 to 303. These lines contain no primes
>> at all.
>
> What you're seeing is almost certainly related to the lines in Ulam's
> Spiral.
> I'm sure there's a Visualisation section in the Prime Links part of
> http://www.primepages.org/

Phil probably thought you were plotting numbers in a spiral but that is not
the case.
You are plotting them like this (possibly mirrored/rotated and I guess you
start at 2):
2 3 4 5 6 7
8 9 10 11 12 13
14 15 16 17 18 19
....
This example is 6 across. Yours is 546 = 2*3*7*13. This means your column
with x at the top contains numbers of the form x + 546*n = x + 2*3*7*13*n.
If any one of 2, 3, 7, 13 divides x then it will divide all numbers of form
x + 2*3*7*13*n.

I guess you start at 2 because that would mean your prime-free columns 241
to 249 and 295 to 303 correspond to x = 242 to 250 and x = 296 to 304. These
x values are all divisible by 2, 3, 7 or 13, so there will never be primes
in those columns.

x and a are called relatively prime if there is no prime which divides both
of them.
x + a*n for fixed x and a with n running through integers is called an
arithmetic progression with common difference a.
The x values of your prime-free columns are those x which are *not*
relatively prime to the common difference a (546 in your case).

Dirichlet's theorem says that if x and a *are* relatively prime then there
are infinitely many primes in the arithmetic progression x + a*n.
So the number of primes in your columns is either 1 (for x = 2, 3, 7, 13) or
0 (for other x divisible by 2, 3, 7 or 13), or infinite (for x not divisible
by 2, 3, 7 or 13).

p#, called p primorial, is the product of all prime numbers <= p. You can
try making grids which are a primorial across, for example 210 = 2*3*5*7.
This gives especially many prime-free columns.

--
Jens Kruse Andersen
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