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Math Help - σ sum of divisors proof

  1. #1
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    σ sum of divisors proof

    σ is an odd integer if and only if n is a perfecdt square or twice a perfect square.

    [Hint: If p i s an odd prime, then 1+p+p^2+...+p^k is odd only when k is even]
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  2. #2
    Super Member Bacterius's Avatar
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    Could you please explain the link between σ, n and p ? I take that σ is the sum of the divisors of n, is that right ?
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  3. #3
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    Quote Originally Posted by MichaelG View Post
    σ is an odd integer if and only if n is a perfecdt square or twice a perfect square.

    [Hint: If p i s an odd prime, then 1+p+p^2+...+p^k is odd only when k is even]

    It is a basic fact that if n=p_1^{a_1}\cdot \ldots \cdot p_k^{a_k}\in\mathbb{N} is the prime decomposition of a natural number, then \sigma(n)=\prod\limits_{i=1}^k\frac{p_i^{a_i+1}-1}{p_i-1}=\prod\limits_{i=1}^k\left(\sum\limits_{m=0}^{a_  i}p_i^m\right)

    Well, now just use the hint...

    Tonio
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