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Re: theoretical question

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  • 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 1 of 4 , Mar 3, 2004
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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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