Some Number Theory Definitions, Identities and Theorems

Following are some definitions, identities, and theorems that arise in number theory complete with the link of each proof/explanation.

1. The Chinese Reminder Theorem: Let m₁, m₂, ..., m_r be pairwise relatively prime positive integers. Then the system of congruence

 

x ≡ a₁ (mod m₁)

x ≡ a₂ (mod m₂)

.

.

.

x ≡ a_r (mod m_r) 

has a unique solution modulo M = m₁m₂...m_r. [Explanation and Proof]

2. The product of greatest common divisor and least common multiple of two positive integers is the product of the two positve integers.

a×b = gcd(a,b)×lcm(a,b).[Explanation Video]

3. For any integer a,b,c,d, and positive integer n, if a ≡ c (mod n) and b ≡ d (mod n) then

a + b ≡ c + d (mod n)[Read More]

4. Euclidean Algorithm: For every integer a and b with a > b,

gcd(a,b) = gcd(b, a mod b). [Explanation] [Calculator]

5. The product of greatest common divisor and least common multiple of two positive integers is the product of the two positve integers.

a×b = gcd(a,b)×lcm(a,b).[Explanation Video]

6. (2m)!(2n)! is a multiple of m!n!(m+n)! for any non-negative integers m and n. [Proof]

7. Euclidean Algorithm: For every integer a and b with a > b,

gcd(a,b) = gcd(b, a mod b). [Explanation] [Calculator]

8. The Chinese Reminder Theorem: Let m₁, m₂, ..., m_r be pairwise relatively prime positive integers. Then the system of congruence

 

x ≡ a₁ (mod m₁)

x ≡ a₂ (mod m₂)

.

.

.

x ≡ a_r (mod m_r) 

has a unique solution modulo M = m₁m₂...m_r. [Explanation and Proof]

9. For any integer a,b,c,d, and positive integer n, if a ≡ c (mod n) and b ≡ d (mod n) then

a + b ≡ c + d (mod n)[Read More]

10. Artin’s Reciprocity Theorem: If R is a number field and R’ a finite integral extension, then there is a surjection from the group of fractional ideals prime to the discriminant, given by the Artin symbol. For some cycle c, the kernel of this surjection contains each principal fractional ideal generated by an element congruent to 1 mod c. [Explanation]