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Math Help - Number Theory - Multiplicative Functions

  1. #1
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    Number Theory - Multiplicative Functions

    Another one of those interesting number theory questions:


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  2. #2
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    Quote Originally Posted by moolimanj View Post
    Another one of those interesting number theory questions:


    F(7)=5
    F(8)=0
    F(9)=3
    F(12)=0.
    Is it right?

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    Now if n is even, say 6 then 1,2,3,4,5,6 any two adjacent ones you take we have one is even and one is odd. Thus \gcd (a,n) , \gcd (a+1,n) cannot be both 1 because at least one of them is at least 2.

    For the converse consider 1,2,...,n and pick the first two and so \gcd(1,n) , \gcd(2,n) we know that \gcd(1,n)=1 but since F(n)=0 it means \gcd (2,n) \not = 1 (otherwise F(n)\not = 0) and so it has to be \gcd(2,n) = 2 which means 2|n.
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    If prime p then in the integers 1,2,...,p we see that \gcd(1,p)=\gcd(2,p) = 1 , \gcd(2,p)=\gcd(3,p) = 1, ... ,\gcd(p-2,p)=\gcd(p-1,p)=1. Thus we have p-2 cases.
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