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Math Help - Finding a conjecture relating prime numbers to divisors

  1. #1
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    Redcar
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    Finding a conjecture relating prime numbers to divisors

    Hello I was just wondering if anyone could help me find a conjecture, theorem, hypothesis around this particular result.

    If p,q are prime numbers, m,n are integers, and f(r) is the number of nontrivial divisors of r (ie. not 1 or r itself).

    That if r=(p^m)*(q^n) (p≠q) that f(r)=(m+1)n+(m-1)

    Thanks, I just really suck at looking for previous groundwork involving this.
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  2. #2
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    Re: Finding a conjecture relating prime numbers to divisors

    Since p and q are unique primes, every factor of r must be a product of some number of p's and some number of q's. A given factor of r can have anywhere between 0 p's in it and m p's in it. Likewise, it can have between 0 q's and n q's.

    You can either use some combinatoric methods or write it out in general in a table and count them up. Consider (m,n) to denote the number of M's and number of N's in a factor. Then one factor is (0,1), which respresents p^0q^1. Another is (0,2). Can you count them all up from there?
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