PRIME BEAUTY {SELF-MADE}
- Phlembac Adib Hasan
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$P$ is a positive integer.Prove that $p=1,4$ or any odd prime if and only if
\[({\prod_{k=0} ^{p-1}{^{p-1}}C_k})^2\equiv1(mod p)\]
\[({\prod_{k=0} ^{p-1}{^{p-1}}C_k})^2\equiv1(mod p)\]
Last edited by Phlembac Adib Hasan on Sun Dec 18, 2011 7:27 pm, edited 2 times in total.
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Re: PRIME BEAUTY {SELF-MADE}
Hint:
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Nur Muhammad Shafiullah | Mahi
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- Phlembac Adib Hasan
- Posts:1016
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Re: PRIME BEAUTY {SELF-MADE}
Sorry, I was an idiot; used $\sum$ notation instead of $\Pi$. However, my problem was correct!
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Re: PRIME BEAUTY {SELF-MADE}
Prove the following very useful lemma.Phlembac Adib Hasan wrote:$P$ is a positive integer.Prove that $p=1,4$ or any odd prime if and only if
\[({\prod_{k=0} ^{p-1}{^{p-1}}C_k})^2\equiv1(mod p)\]
\[\binom{p-1}k\equiv(-1)^k\pmod p\]
Then just square.
One one thing is neutral in the universe, that is $0$.
Re: PRIME BEAUTY {SELF-MADE}
What was this called? I forgot the ame... :/
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Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Re: PRIME BEAUTY {SELF-MADE}
This lemma has no name in fact.
One one thing is neutral in the universe, that is $0$.
- zadid xcalibured
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Re: PRIME BEAUTY {SELF-MADE}
cant we name it phlembac lemma?
- afif mansib ch
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Re: PRIME BEAUTY {SELF-MADE}
\[(p-1)(p-2)..(p-k)/k!\equiv -1.-2....-k/1.2....k(modp)\]
or\[\binom{p-1}{k}^2\equiv 1(mod p)\]
so the multiplication will also be congruent mod p.isn't it enough?
or\[\binom{p-1}{k}^2\equiv 1(mod p)\]
so the multiplication will also be congruent mod p.isn't it enough?
Re: PRIME BEAUTY {SELF-MADE}
Yes, it will.afif mansib ch wrote:\[(p-1)(p-2)..(p-k)/k!\equiv -1.-2....-k/1.2....k(modp)\]
or\[\binom{p-1}{k}^2\equiv 1(mod p)\]
so the multiplication will also be congruent mod p.isn't it enough?
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Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
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