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Thread: prime-power

  1. #1
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    Post prime-power

    1) Prove that if n is a square, then each exponent in its prime-power decomposition is even.
    and
    2) Prove that if each exponent in the prime-power decomposition of n is even, then n is square.
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  2. #2
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    Quote Originally Posted by Sally_Math View Post
    1) Prove that if n is a square, then each exponent in its prime-power decomposition is even.
    and
    I do the first one and leave the second one for thee to think about.

    (Assuming $\displaystyle n>1$)

    If $\displaystyle n$ is a square it means $\displaystyle n=m^2$. Now $\displaystyle m>1$ and so it can be written as $\displaystyle m = p_1^{a_1}...p_k^{a_k}$ by prime decomposition. This means, $\displaystyle n = \left( p_1^{a_1}...p_k^{a_k} \right)^2 = p_1^{2a_1} ... p_k^{2a_k}$. And so exponents in prime decomposition of $\displaystyle n$ are even.

    Similar problem: Once you prove the above problem try proving a stronger result. That $\displaystyle n>1$ is an $\displaystyle m$-th power if and only if each prime in the decomposition of $\displaystyle n$ is a multiple of $\displaystyle m$.
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