Difference between revisions of "Proofs to Some Number Theory Facts"

Latest revision as of 13:47, 16 May 2025

There are some very useful facts in Number Theory that have no names. If you have a fact, feel free to add it to this page.

For a prime number $p$ , we have

$\dbinom{2p}{p} \equiv 2 \pmod {p}$

We have the congruence

$(p-1)! \cdot \dbinom{2p}{p} = 2 \cdot (2p-1) \cdot (2p-2) \cdot \dots \cdot (p+1) \equiv 2 \cdot (p-1)! \equiv -2 \pmod {p}$

$\implies \dbinom{2p}{p} \equiv 2 \pmod {p}$

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-There are some very useful facts in [[Number Theory]] that have no names.
+There are some very useful facts in [[Number Theory]] that have no names. If you have a fact, feel free to add it to this page.
 ==Fact 1==
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 <cmath>\implies \dbinom{2p}{p} \equiv 2 \pmod {p}</cmath>
-===Uses===
-===Examples===
+== See Also ==
-===Problems===
+* [[Number Theory]]
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-==Fact 4==
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-==Fact 7==
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-==See Also==
-*[[Number Theory]]
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