Not Necessarily Distinct
- 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:
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
Here's an illustration of that from my web page:
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.