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Homework Help: Hermite identity help

  1. Jun 16, 2013 #1
    1. The problem statement, all variables and given/known data

    Let a and b be integers and m an integer >1 Evaluate

    [b/m] + [(b+a)/m]+ [(b+2a)/m]+ [(b+3a)/m]+ [(b+4a)/m]+ [(b+5a)/m]+.....+ [(b+(m-1)a)/m]

    2. Relevant equations

    3. The attempt at a solution
    i tried to use hermite identity.

    [x] + [x + 1/n] + [x + 2/n] +...+ [x + (n-1)/n] = [nx]

    assuming x = b/m and 1/n = a/m.. but a/m is not an integer so i cant use it. I m stuck what to do?
  2. jcsd
  3. Jun 17, 2013 #2


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    Homework Helper

    Funny you should mention that, just last week I read about it in pp 90-94 of Concrete Mathematics by Ronald L. Graham , Donald E. Knuth, and Oren Patashnik. Several interesting things are the dependence of the result on gcd(a,m) ,the fact that

    $$\sum_{0 \le k < m} \left[ \frac{b+k \, a}{m}\right] = \sum_{0 \le k < a} \left[ \frac{b+k \, m}{a}\right], \, \, \, \, \, \, \, \, \, \mathop{integers \, \, \, a,m>0} $$

    the closely related integrals

    $$\frac{1}{m} \int_0^m (a \, x+b) \! \mathop{dx}=\frac{1}{a} \int_0^a (m \, x+b) \! \mathop{dx}=\frac{a \, m}{2}+b$$

    The book uses several special cases to deduce the general one. Give it another try. If you have trouble describe the methods you tried and those you have know.
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