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 [number theory] sieving with a_i mod p_i jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 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 « Next Oldest | Next Newest »

 Messages In This Thread [number theory] sieving with a_i mod p_i - by tommy1729 - 09/09/2014, 01:10 AM RE: [number theory] sieving with a_i mod p_i - by jaydfox - 09/10/2014, 04:50 PM RE: [number theory] sieving with a_i mod p_i - by tommy1729 - 09/10/2014, 08:27 PM RE: [number theory] sieving with a_i mod p_i - by jaydfox - 09/11/2014, 01:47 AM RE: [number theory] sieving with a_i mod p_i - by tommy1729 - 09/11/2014, 08:24 AM RE: [number theory] sieving with a_i mod p_i - by jaydfox - 09/11/2014, 08:00 PM RE: [number theory] sieving with a_i mod p_i - by tommy1729 - 09/12/2014, 07:28 AM RE: [number theory] sieving with a_i mod p_i - by tommy1729 - 09/11/2014, 11:40 AM

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