1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Approximate polynomials as frequencies

  1. Feb 23, 2008 #1
    I want to define the frequency, p(n), to be the amount of times (in an integral domain) where a function happens to cube.

    For instance,

    f(x) = x^2 happens to be a cube when (x=1, x= 8, x = 27, x = 64), the frequency is thus 4/64 = 1/16 and happens to follow the relation [(n^(0.333333)]/(n) = p(n), where n is the number of integers in the domain, and p(n) is the frequency. The frequency function doesnt seem to change with an increasing n.

    however a small tweak to the function can actually have profound effects on the p(n)
    value and the adding of an extra term can make it much more complex.

    For instance,

    f(x) = x^2 + x^3 = y^3 (where x and y are integers)

    and we can actually create an upper bound for the frequency, not a good one, but better than 1. If x^2 is a cube, then it goes against fermat's last theorem, and we can eliminate all these possibilities as such,

    (1- ((n^0.3333333)/n) > p(n)

    a lower bound is not so obvious but can also be constructed i would guess, possibly because the function has little to share intersections with the fermat's last theorem eqn. of course, now it becomes about probability.

    any ideas?
    Last edited: Feb 24, 2008
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted