what is this pattern in the primes?

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Hi,

What is the meaning of this pattern:

Primorial primefactors coprimes
2 2
6 2x3 2
30 6x5 8 ((5-1)*2)
210 30x7 48 ((7-1)*8)
2310 210x11 480 ((11-1)*48)


There is the primes-1 sequence in there for the coprimes formula,
ie to get EulerPhi(2310)48%0 it is (11-1)*48

Question: how do you explain coprimes being
(nextprimefactor-1)*(coprimes of next smallest primorial)?

cheers,
Jamie


Re: what is this pattern in the primes?
On 25/03/2016 5:07 PM, Jamie M wrote:
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Write the numbers from 1 to your current primorial. If that's 6, then  
you have

1 2 3 4 5 6

Now remove the numbers that are divisible by the factors

1 X X X 5 X

Now consider your next primorial, which is given by multiplying the  
current by the next prime, p.

Create p - 1 new lines by adding your current primorial to each number,  
successively:

  1 X X X  5 X
  7 X X X 11 X
13 X X X 17 X
19 X X X 23 X
25 X X X 29 X

Now, this contains all the numbers from 1 to the new primorial, with the  
multiples of the current primorial's factors already removed.

Since the previous primorial is coprime with the new factor p, each  
column that hasn't already been removed contains exactly one number that  
is divisible by p. We can rearrange the columns to put that number last,  
then remove the entirety of the last row.

  1 X X X 11 X
  7 X X X 17 X
13 X X X 23 X
19 X X X 29 X

So the count of numbers remaining (being the coprime count for the next  
primorial) is (p-1) * the count of numbers in the first row (being the  
coprime count for the current primorial).

Sylvia.









Re: what is this pattern in the primes?
On 3/25/2016 5:36 AM, Sylvia Else wrote:
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Hi,

Thanks, seems to make sense.

I noticed this just now too:

primorial 210 is the smallest primorial that has more coprimes than the
count of primes smaller than it (48 coprimes vs 46 primes smaller than
210), then for larger primorials the two numbers converge I think at
infinity.

primorial coprimes countOfPrimesLessThanPrimorial ratio
2 1 0 0
6 2 3 1.5
30 8 10 1.25
210 48 46 0.9583...
2310 480 343 0.7146...
30030 5760 3248 0.5638...
510510 92160 42331 0.4593...

If the count of primes and coprimes for a primorial converges
at infinity, then the formula:

coprimes=(nextprimefactor-1)*(coprimes of next smallest primorial)

should give an accurate count of how many primes are below an infinite
sized primorial as it is the same as the number of coprimes.

Also it would be interesting to find the error term between
the count of primes below a primorial and the count of the
primorial's coprimes for smaller primorials, to get an
accurate count of primes in different ranges of numbers.

cheers,
Jamie








Re: what is this pattern in the primes?
On 3/25/2016 1:01 PM, Jamie M wrote:
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I just checked and the sequence 0,3,10,46,343...
is in oeis.org:

http://oeis.org/A000849

"Number of primes <= product of first n primes"

(product of first n primes is a primorial)

0, 1, 3, 10, 46, 343, 3248, 42331, 646029, 12283531, 300369796,  
8028643010, 259488750744, 9414916809095, 362597750396740,  
15397728527812858, 742238179058722891, 40068968501510691894

cheers,
Jamie


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Re: what is this pattern in the primes?
On 3/25/2016 1:07 PM, Jamie M wrote:

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Linked from the oeis page:

http://mathematical.com/mathprimorial1.html

Proof for determining the number of primes below
a given number using primorials I think.

cheers,
Jamie

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Re: what is this pattern in the primes?
On 26/03/2016 7:01 AM, Jamie M wrote:
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Your ratio is (primes less than n) / (coprimes of n less than n). It is  
reducing, which is consistent with my intuition that the coprimes would  
rise faster than primes. After all, each higher primordial loses exactly  
one prime as a coprime, but gets a bunch of other primes as coprimes.

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Given the above - not at all.

Sylvia.


Re: what is this pattern in the primes?
On 3/25/2016 7:15 PM, Sylvia Else wrote:
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Hi,

Thanks for the correction, let me restate it:

for all numbers below an infinite primorial,
the number of primes are infinitely less than
the number of coprimes, and the number of coprimes
is infinitely less than the magnitude of the infinite
primorial.

So there are infinitely many more coprimes than primes,
but at the same time both primes AND coprimes approach
zero density in the natural numbers up to the infinite
primorial.

cheers,
Jamie


Re: what is this pattern in the primes?
On 28/03/2016 7:41 AM, Jamie M wrote:

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Your really need to stop referring to an infinite promorial. There is no  
such thing. Your propositions are better expressed as

(primes less than n) / (coprimes of n less than n) --> 0, as n--> ?

and

(coprimes of n less than n) / n --> 0, as n--> ?

However, neither proposition has been proved here. I supported my  
intuition by means of a hand-waving argument, but that's all it was. It  
wasn't a proof.

Sylvia.


Re: what is this pattern in the primes?
On 3/27/2016 5:44 PM, Sylvia Else wrote:
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No that is a totally different statement and not true, since
the number of coprimes less than n varies up and down depending on n,
it isn't a non oscillating relationship unless n values are restricted
to certain classes of numbers, ie primorials (or factorials too
probably).

An Infinite primorial concept is no big deal it is just the product
of all primes, I specifically stated it for primorials only and not
for any n.

cheers,
Jamie



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