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

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<cmath>\implies \dbinom{2p}{p} \equiv 2 \pmod {p}</cmath>
 
<cmath>\implies \dbinom{2p}{p} \equiv 2 \pmod {p}</cmath>
  
===Uses===
 
  
===Examples===
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== See Also ==
  
===Problems===
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* [[Number Theory]]
 
 
==Fact 2==
 
 
 
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==Fact 6==
 
 
 
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*[[Number Theory]]
 
  
 
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Revision as of 16:47, 28 April 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.

Fact 1

Statement

For a prime number $p$, we have

\[\dbinom{2p}{p} \equiv 2 \pmod {p}\]

Proof

We have the congruence

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

\[\implies \dbinom{2p}{p} \equiv 2 \pmod {p}\]


See Also

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