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Thread: Proving on positive integers?

  1. #1
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    Proving on positive integers?

    Prove that for all positive integers a and b, σ(ab) ≤ σ (a) σ (b).
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  2. #2
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    Re: Proving on positive integers?

    Is this the sum of divisors function?
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  3. #3
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    Re: Proving on positive integers?

    The sum of divisors function $\displaystyle \sigma$ is multiplicative for relatively prime integers. We will consider any common prime factor (p) of a and b. Suppose that $\displaystyle p^k$ completely divides $\displaystyle a$ and $\displaystyle p^m$ completely divides $\displaystyle b$.

    You want to prove that

    $\displaystyle \sigma(p^{k+m}) \le \sigma (p^k) \sigma (p^m)$

    However, finding the sum of divisors of a prime power is easy (just a geometric series).

    $\displaystyle 1 + p + \dots + p^{k+m} \le (1 + p + \dots p^k)(1 + p + \dots p^m)$

    And you can go on from there.
    Last edited by richard1234; Aug 21st 2012 at 10:14 AM.
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