Proofs to Some Number Theory Facts
There are some very useful facts in Number Theory that have no names.
Fact 1
Statement
For a prime number 
, we have
![\[\dbinom{2p}{p} \equiv 2 \pmod {p}\]](http://latex.artofproblemsolving.com/0/d/5/0d541009345d1f52ce54824d887112253e498d7f.png)
Proof
We have the congruence
![\[(p-1)! \cdot \dbinom{2p}{p} = 2 \cdot (2p-1) \cdot (2p-2) \cdot \dots \cdot (p+1) \equiv 2 \cdot (p-1)! \equiv -2 \pmod {p}\]](http://latex.artofproblemsolving.com/a/2/d/a2d55a204aa2adb8277cc6250d6b1c90f4c53d39.png)
![\[\implies \dbinom{2p}{p} \equiv 2 \pmod {p}\]](http://latex.artofproblemsolving.com/b/4/4/b443791c19643c53e6120edf6c0fdfa4de51f754.png)
Uses
Examples
Problems
Fact 2
Statement
Proof
Uses
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Problems
Fact 3
Statement
Proof
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Examples
Problems
Fact 4
Statement
Proof
Uses
Examples
Problems
Fact 5
Statement
Proof
Uses
Examples
Problems
Fact 6
Statement
Proof
Uses
Examples
Problems
Fact 7
Statement
Proof
Uses
Examples
Problems
See Also
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