Difference between revisions of "Proofs to Some Number Theory Facts"
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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== | ==Fact 1== | ||
===Statement=== | ===Statement=== | ||
| + | For a prime number <math>p</math>, we have | ||
| + | |||
| + | <cmath>\dbinom{2p}{p} \equiv 2 \pmod {p}</cmath> | ||
===Proof=== | ===Proof=== | ||
| + | We have the congruence | ||
| + | |||
| + | <cmath>(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}</cmath> | ||
| + | |||
| + | <cmath>\implies \dbinom{2p}{p} \equiv 2 \pmod {p}</cmath> | ||
===Uses=== | ===Uses=== | ||
| Line 12: | Line 20: | ||
===Problems=== | ===Problems=== | ||
| − | |||
==Fact 2== | ==Fact 2== | ||
Latest revision as of 22:03, 9 January 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 
, 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
Examples
Problems
Fact 3
Statement
Proof
Uses
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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