Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
[number theory] sieving with a_i mod p_i
#2
(09/09/2014, 01:10 AM)tommy1729 Wrote: Consider all the primes between 1 and 100.
Call them p_i.

If we want to count the primes between 100 and 10000 then we sieve that interval with 0 mod p_i.

But what happens if say we sieve 7 mod p_i ?
( 7 mod 2 => 1 mod 2 , 7 mod 3 => 1 mod 3 , 7 mod 5 => 2 mod 5 , 7 mod 7 => 0 mod 7 , 7 mod 11 , ... )

In general what happens if we sieve a_i mod p_i with 0 < a_i < p_i ?


How do we choose the a_i such that we sieve as many numbers as possible ?
Or how do we choose the a_i such that there are as many numbers left as possible ?

regards

tommy1729

It took me a couple minutes to really figure out what you were even asking here. Once I did, I threw together some tests in Excel. I was expecting it to be very erratic, but counter-intuitively, it's really stable. I haven't had a chance to validate these numbers or write some gp code, so these numbers are very preliminary, but...

Taking primes between 2 and 97 ("100"), and sieving the numbers from 101 to 10000, I get:

Code:
a_i     Count("Primes")
0..27   1204
28..33  1203
34..51  1202
52..59  1201
60..69  1200
70..71  1199
72..77  1198
78..93  1197
94..96  1196

I suppose the numbers might be more stable if I had sieved from 98 to 97^2? Or perhaps from 98 to 101^2-1? Anyway, this single data point suggests that low numbers, close to 0, maximize the number of "primes".
~ Jay Daniel Fox
Reply


Messages In This Thread
RE: [number theory] sieving with a_i mod p_i - by jaydfox - 09/10/2014, 04:50 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Hyper operators in computability theory JmsNxn 5 5,111 02/15/2017, 10:07 PM
Last Post: MphLee
  Cellular auto : rule 30 number ? tommy1729 0 1,559 08/03/2016, 08:31 PM
Last Post: tommy1729
  Set theory debate : cantor 1st / Virgil argument. tommy1729 1 2,348 12/08/2015, 11:14 PM
Last Post: tommy1729
  [2015] Spiderweb theory tommy1729 0 2,115 03/29/2015, 06:25 PM
Last Post: tommy1729
  " fake ring theory " tommy1729 0 2,249 06/11/2014, 11:29 PM
Last Post: tommy1729
  [Number Theory] pi(X,x,x+2)+pi(X,x,x+4) tommy1729 1 2,725 04/11/2014, 10:33 PM
Last Post: tommy1729
  the " average restart " of a number. tommy1729 0 1,800 03/26/2014, 01:02 AM
Last Post: tommy1729
  A conjecture about number theory and tetration tommy1729 0 2,229 10/23/2013, 09:28 PM
Last Post: tommy1729
  (number theory) How tommy1729 does it. tommy1729 0 2,206 08/12/2013, 09:13 PM
Last Post: tommy1729
  Number theory and hyper operators JmsNxn 7 8,864 05/29/2013, 09:24 PM
Last Post: MphLee



Users browsing this thread: 1 Guest(s)