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The proof of $(n+1)!(n+2)!$ divides $(2n+2)!$ for any positive integer $n$
Does $(n+1)!(n+2)!$ divide $(2n+2)!$ for any positive integer $n$?
I tried to prove this when I was trying to prove the fact that ${P_n}^4$ divides $P_{2n}$ where $n$ is a positive integer, where $P_{...
- 428
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Logarithm of factorial equal to sum of logarithm of primes
Let $N$ a positive integer. Denote $\mathcal{P}$ the set of prime numbers. I have to show that
\begin{align}
\log(N!) = \sum_{p^{\nu}\leq N \\ p\in \mathcal{P}} \left\lfloor\dfrac{N}{p^{\nu}}\right\...
- 866