PRIME BEAUTY {SELF-MADE}

[Phlembac Adib Hasan]

$P$ is a positive integer.Prove that $p=1,4$ or any odd prime if and only if
\[({\prod_{k=0} ^{p-1}{^{p-1}}C_k})^2\equiv1(mod p)\]

[*Mahi*]

Hint:

Binomial theorem and FLT

[Phlembac Adib Hasan]

Sorry, I was an idiot; used $\sum$ notation instead of $\Pi$. However, my problem was correct!

[Masum]

$P$ is a positive integer.Prove that $p=1,4$ or any odd prime if and only if
\[({\prod_{k=0} ^{p-1}{^{p-1}}C_k})^2\equiv1(mod p)\]

Prove the following very useful lemma.
\[\binom{p-1}k\equiv(-1)^k\pmod p\]
Then just square.

[*Mahi*]

What was this called? I forgot the ame... :/

[Masum]

This lemma has no name in fact.

[zadid xcalibured]

cant we name it phlembac lemma?

[Masum]

I wonder why

[afif mansib ch]

\[(p-1)(p-2)..(p-k)/k!\equiv -1.-2....-k/1.2....k(modp)\]
or\[\binom{p-1}{k}^2\equiv 1(mod p)\]
so the multiplication will also be congruent mod p.isn't it enough?

[*Mahi*]

\[(p-1)(p-2)..(p-k)/k!\equiv -1.-2....-k/1.2....k(modp)\]
or\[\binom{p-1}{k}^2\equiv 1(mod p)\]
so the multiplication will also be congruent mod p.isn't it enough?

Yes, it will.