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

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

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

@@ Line 1: / Line 1: @@
-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==
 ===Statement===
+For a prime number <math>p</math>, we have
-===Proof===
+<cmath>\dbinom{2p}{p} \equiv 2 \pmod {p}</cmath>
-===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===
+We have the congruence
-===Uses===
+<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>
-===Examples===
+<cmath>\implies \dbinom{2p}{p} \equiv 2 \pmod {p}</cmath>
-===Problems===
-==See Also==
+== See Also ==
-*[[Number Theory]]
+* [[Number Theory]]
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