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Not Necessarily Distinct

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  • joel.levenson
    I don t know if any of you remember, but a few months ago I was asking some rather inept questions here about greatest prime factors and some sets and
    Message 1 of 1 , Apr 26, 2007
      I don't know if any of you remember, but a few months ago I was asking
      some rather inept questions here about "greatest prime factors" and
      some "sets" and "patterns" that were baffling me. I was trying to
      make sense of an observation about a particularly cool way (imho) in
      which the integers can be ordered in terms of the prime numbers. I've
      managed to pin it down, I think, and express it fairly coherently, for
      whatever its worth.

      In one sentence: it's a way the integers can be arranged according to
      the binomial expansion and the sequence of prime numbers.

      I've given as good an account as I can of what that sentence is
      supposed to mean on a web page at:

      http://www.individual.utoronto.ca/levenson/nnd.htm

      Here's a brief summary: The integers can be divided into sets whose
      sizes are the binomial coefficients. The sets are also defined by the
      integers' greatest prime factors, so the they are also defined by the
      sequence of prime numbers.

      For me, the neatest way to see what this looks like is to superimpose
      the sets on Pascal's triangle. You get the primes running in order
      down one side and the powers of two running down the other (with two
      at the top, because it is both the first prime and two to the first
      power.)

      Here's an illustration of that from my web page:

      http://individual.utoronto.ca/levenson/nnd.htm#ill3

      I want to mention all the people here who gave me advice and help last
      time I was here: David Broadhurst, Phil Carmody, Payam Samidoost,
      Joshua Zucker, Chris Caldwell, and Kermit Rose. I was quite stuck,
      and y'all helped unstick me.
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