09/10/2014, 04:50 PM

(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