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I need an elementary proof of the fact(that is without using Bertrand's Postulate or something like that): Let $p(n)$ be the largest prime not greater than $n$. Then $p(n)$ will come exactly once in the factorization of $n!$.

Corollary: $n!$ is not a perfect power.

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    Do you know if such an elementary proof exists or you just think there is? Or wished...2011-07-21
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    Applying your fact to even $n$ shows that the largest prime no greater than $n$ is larger than $n/2$. In other words, proving the fact is as hard as proving Bertrand's postulate.2011-07-21
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    @ccc, why not post that as an answer?2011-07-21
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    Related: http://math.stackexchange.com/questions/31973/n-is-never-a-perfect-square-if-n-geq2-is-there-a-proof-of-this-that-doesnt2011-07-21
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    @ccc: Gerry's right; do post an answer.2011-10-04

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