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Greatest common factor with exponents of first input

  1. Jan 13, 2016 #1
    ok so is there a function that exists (for all intents and purposes let's call it G(x,y) )where

    x= a^2*b^4*c
    y=a^4*b^2*d

    G(x,y) = a^2*b^4

    basically gcd, but the exponents match those of the common prime factors of the first input (x)

    ********
    equally useful would be a function where the output would be the coprime factor(s) of the first input. ie
    B(x,y) = c
     
  2. jcsd
  3. Jan 14, 2016 #2

    Mark44

    Staff: Mentor

    Actually the result would be ##a^2b^2##.

    A better way to write this would be ##G(a^2b^4c, a^4b^2d) = a^2b^2##.
    Is there such a function? I assume your question is, "Is there a function in some programming language that returns the GCF of its two inputs?"
    I don't know, but possibly someone has written such a library function in, say, the numpy or scipy Python libraries, or some other library in some other programming language.


     
  4. Jan 14, 2016 #3
    ok let me be more specific:

    first off i am looking for a mathematical algorithm, something that could exist outside of a computer program

    secondly, the equation given was purely just an example. What I need is an algorithm "G(x,y)" where, given two functions (a first function "x" and a second function "y"), G(x,y) returns a number who has all and only prime factors that are the shared prime factors of x and y, but are taken to the greatest exponents that still divide x (the first of the two functions plugged into G)

    alternatively, equally useful to me would be an algorithm "B(x,y)" where, given the same two functions x and y, B(x,y) returns a number that is divisible by all and only the prime factors of x (with the exponents that they have as divisors of x) that are not shared with y.
     
  5. Jan 14, 2016 #4

    Mark44

    Staff: Mentor

    I'm not aware of anything like this. Are you saying that, for example, ##G(2^{10}, 2) = 2^{10}##?
    I don't know how useful such an algorithm would be, but all an algorithm is, is a sequence of steps to be performed. Assuming my example follows what you're describing, how would you explain to someone how a result is achieved? That would be your algorithm.
     
    Last edited: Jan 14, 2016
  6. Jan 14, 2016 #5
    Yeah exactly: I'm looking for a function G(x,y) that would work exactly like the example you gave. G(2^10, 2) = 2^10.

    The Algorithm would be extremely useful to me personally. I'm looking for as nice and clean and simple a function as possible. The best example i can think of so far would be G(x,y) = gcd(x,y^x), but i'd prefer something more precise, and something that would not require processing unnecessarily large numbers, as y^x would get very very large.

    If such a G(x,y) can not be easily and cleanly created, a function that would be of equal use to me would be a function B(x,y) that would return the divisors of x that are coprime to y. For example B(2^10*3^5*5^8*7^3*11, 2^3*3^2*7^2) = 5^8*11



    If it is at all helpful, x will always be a factorial and y will always be made up of factors less than or equal to x. So x will have prime factors that y won't have, but y will only be made up of prime factors that x also has. Also the exponents on the prime factors of y will *probably* always be less then the exponents of the prime factors of x.
     
    Last edited by a moderator: Jan 14, 2016
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