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Re: [PrimeNumbers] theoretical question

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  • mikeoakes2@aol.com
    I wrote ... For some reason I had not seen Jens s post, which is (a) neater and (b) proves the general case. So I am erasing my post - sorry for wasting your
    Message 1 of 4 , Mar 3 1:16 AM
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      I wrote
      >Probably, this argument can be extended, by induction, to derive the same
      >affirmative conclusion for any (finite) number of primes.

      For some reason I had not seen Jens's post, which is (a) neater and (b)
      proves the general case.
      So I am erasing my post - sorry for wasting your time.

      Mike


      [Non-text portions of this message have been removed]
    • asposer
      Thanks for your reply, I ve read about the Chinese Remainder Theorem but I ve struggled with understanding its implications. I m also interested in functions
      Message 2 of 4 , Mar 3 5:44 PM
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        Thanks for your reply, I've read about the Chinese Remainder Theorem
        but I've struggled with understanding its implications.

        I'm also interested in functions which produce as their output the
        positive integers or lists of integers whose composition (with
        multiplicity) follows that of the positive integers. In this case,
        finding difference between successive pairs of prime/integer which
        satisfy the requirements and the starting term produces the sequence
        3, 6, 9, 12, etc. (which reduces to 1, 2, 3, 4, etc.) and Omega(n) of
        the resulting sequence is equal to Omega(n) + 1 of the positive
        integers. If anyone happens to know where I can find more info about
        functions with these properties, I'd appreciate a pointer in the
        right direction (books, links, etc). Thanks!

        -Andrew-


        --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
        <jens.k.a@g...> wrote:
        > asposer wrote:
        > > Desired n: 3
        > >
        > > 5 * 12 = 60
        > > 7 * 9 = 63
        > > 11 * 6 = 66
        > > 3 * 23 = 69
        >
        > This is always possible for any n and any order of the primes.
        > The problem is just solving a set of modular equations.
        > Your example, with x for the first product, is the smallest
        solution to:
        > x = 0 (mod 5)
        > x = -3 (mod 7)
        > x = -6 (mod 11)
        > x = -9 (mod 3)
        > CRT (Chinese Remainder Theorem) says it can always be solved when
        the numbers
        > are relatively prime, and primes are that.
        >
        > Setting up such a system of modular equations with carefully chosen
        residues
        > (not k*n) is a great way to find large prime gaps, by ensuring many
        numbers with
        > a small factor.
        >
        > > is it possible to do this with
        > > an infinitely large list of primes?
        >
        > No. A solution must have given finite values for x and n. The
        average prime gap
        > tends to infinite, so infinitely many of the primes will be larger
        than the
        > number they would have to divide.
        >
        > --
        > Jens Kruse Andersen
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