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Prove that the equation is satisfied at least once

  1. Oct 15, 2017 #1
    1. The problem statement, all variables and given/known data
    f(n) is function that takes input n and outputs the smallest integer grater that n^pi
    prove that there exists natural numbers abcdwxyz that are all not smaller than 2015 such that equation is satisfied
    f(x) + f(y) + f(z) + f(w) = f(a) + f(b) + f(c) + f(d)
    and they abcd and wxyz are not trivial meaning that a,b,c,d is not equal to w,x,y,z

    2. Relevant equations

    well ordering principal
    or maybe induction
    3. The attempt at a solution
    f(x) + f(y) + f(z) + f(w) = f(a) + f(b) + f(c) + f(d)
    f(x) = [x^pi + 1]
    f(y) = [y^pi + 1]
    and so on
    [x] means greatest number smaller than x for example [32.23] = 32 [12.99] = 12
    let x0 be the smallest x that satisfies the equation
    and i am stuck i don't know what to do afterwards
  2. jcsd
  3. Oct 15, 2017 #2


    User Avatar
    2017 Award

    Staff: Mentor

    The pigeonhole principle can help here.
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