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Conjecture: $\prod\limits_{k=0}^{n}\binom{2n}{k}$ is divisible by $\prod\limits_{k=0}^n\binom{2k}{k}$ only if $n=1,2,5$.
The diagram shows Pascal's triangle down to row $10$.
I noticed that the product of the blue numbers is divisible by the product of the orange numbers. That is (including the bottom centre number $...
- 25.8k
4
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1
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Proof that each prime power of ${2n \choose n}$ is $\leq \log_p 2n$
I'm trying to work through this proof of the prime number theorem.
Def: Let $P_p(x)$ be the prime power of $p$ in the prime factorization of $x$. I.e. for any natural number $x$, $x = \prod_{p\in \...
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-2
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Divisor Problem [closed]
Express the numbers $11! = 39,916,800$ and the binomial coefficient $\binom {23} {11}$, each as products of their prime factors. Do this without using your calculator in any way. Use this to calculate ...
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Express $11!$ and $\binom{23}{11}$ as products of their prime factors
I'm a bit stuck on how to figure this question out without a calculator and what kind of working I'm supposed to show. Any help would be appreciated, thank you. $\ddot\smile$
Factorise $11!$ and $\...
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