For a program I wrote, I used the property that the power of the highest prime factor of a factorial is always 1. I couldn't find anything about this, but it felt right. I can't prove it. Is my assumption correct or is it just a coincidence?

  • $\begingroup$ What is the "last" prime factor of a factorial? $\endgroup$ – Christoph Oct 16 '15 at 12:28
  • $\begingroup$ I meant the highest. Edited the question. $\endgroup$ – Nico Ekkart Oct 16 '15 at 12:30
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    $\begingroup$ It's implied by the truth of Bertrand's postulate. $\endgroup$ – Daniel Fischer Oct 16 '15 at 12:30
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    $\begingroup$ It's been proven that there is always a prime between $n$ and $2n$ for $n\ge2$. $\endgroup$ – dejongbrent Oct 16 '15 at 12:30
  • $\begingroup$ Ah. Thank you. Didn't know about this. $\endgroup$ – Nico Ekkart Oct 16 '15 at 12:31

If p ≤ n is a prime number, then n! = 1 * 2 * ... * p * ... * n has a prime factor p. And no prime number q > n can be a prime factor of n!

Let p be the largest prime ≤ n. n! has a prime factor p. If p^2 were a factor of n!, then there would have to be two or more numbers ≤ n with a factor p. The smallest two such numbers are p and 2p. So if p^2 were a factor of n!, then 2p ≤ n.

We assumed that p is the largest prime ≤ n, so there is no prime number q with p < q ≤ n. But there is a theorem that there is a prime between p and 2p if p ≥ 2.


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