Number Theory
Number theory, as the name suggests, is the study of the properties and relationships of integers and has a long history dating back to the very inception of mathematics. Gauss famously said about number theory:
“Mathematics is the queen of sciences, and number theory is the queen of mathematics.”
Elementary Number Theory
The term “elementary” in this context contrasts with analytic or algebraic number theory and does not imply simplicity at the level of elementary school. While understanding the propositions may be relatively straightforward, the field itself is not inherently simple. Many mathematically talented individuals are introduced to number theory from a young age.
- Rational Numbers
- Irrational Numbers
Multiples and Divisors
- Greatest Common Divisor and Coprime $\gcd$
- Quotient and Remainder
- Extended Euclidean Theorem
- Divisibility Rules for 3 and 9
- Divisibility Rules for 7 and 13
- Divisibility Rule for 11
Primes
Modular Arithmetic
- Congruence in Number Theory
- Fundamental Theorem of Algebra for Modular Equations
- Fermat’s Little Theorem
- Wilson’s Theorem
- Chinese Remainder Theorem
- Method of Successive Squaring
- Order in Number Theory
Perfect Numbers
- Euler’s Totient Function $\phi$
- Roots of Congruence Equations
- Sigma Function in Number Theory $\sigma$
- Euclid’s Perfect Number Formula
- Euler’s Perfect Number Theorem
Quadratic Reciprocity
- Quadratic Residues and Non-Residues QR, NR
- Multiplicative Property of the Legendre Symbol
- Euler’s Criterion
- Gauss’s Law of Quadratic Reciprocity
- Necessary and Sufficient Condition for a Prime to Leave a Remainder of 1 When Divided by 4
- Necessary and Sufficient Condition for a Prime to Leave a Remainder of 1 When Divided by 3
Elliptic Curves
Cryptography
Initially, cryptography was a prominent application of number theory. For thousands of years, number theory, once purely a branch of pure mathematics, transformed into a practical field through applications in cryptography, and recently, abstract algebra has been applied.
Discrete Logarithms
- Discrete Logarithms
- Diffie-Hellman Key Exchange Algorithm
- ElGamal Public Key Cryptosystem
- Shanks’ Algorithm
- Pohlig-Hellman Algorithm
- Conditions Under Which the Discrete Logarithm Problem Is Easily Solvable
Factorization
- Prime Factorization
- RSA Public Key Cryptosystem
- Goldwasser-Micali Probabilistic Key Cryptosystem
- Pollard’s $p-1$ Factorization Algorithm
- Conditions Under Which Factorization of Semiprimes Is Easily Solvable
Algebraic Number Theory
Extended Rings
- Gaussian Integers $\mathbb{Z} [i]$
- Norm of Gaussian Rings
- Gaussian Prime Theorem
- Eisenstein Integers $\mathbb{Z} [\omega]$
- Norm of Eisenstein Rings
- Eisenstein Prime Theorem
Analytic Number Theory
Arithmetic Functions
- Arithmetic Function $f$
- Multiplicative Property of Arithmetic Functions
- Dirichlet Convolution $\ast$
- Möbius Inversion Formula
- Bell Series $f_{p}(x)$
- Derivative of Arithmetic Functions $f ‘$
- Selberg Identity
- Generalized Dirichlet Convolution $\circ$
References
- Silverman. (2012). A Friendly Introduction to Number Theory (4th Edition)
- Apostol. (1976). Introduction to Analytic Number Theory
- Hoffstein. (2008). An Introduction to Mathematical Cryptography
All posts
- A Simpler Proof of the Divisibility Test for 11
- Division Test for 3 and Proof of the Division Test for 9
- The Proof of the Divisibility Test for 7 and 13
- Euclid's Proof: There are Infinitely Many Primes
- Euclidean Algorithm Proof
- Proof of the Fundamental Theorem of Algebra for Congruent Equations
- Congruences in Number Theory
- Prime Factorization Theorem
- Proof of the Extended Euclidean Theorem
- Euler's Proof: There are Infinitely Many Primes
- Fundamental Theorem of Arithmetic Proof
- One of the Pythagorean Triples Must Be an Even Number
- Pythagorean Triple
- One of the Pythagorean triples must be a multiple of three.
- Primitive Pythagorean Triples Can Be Expressed Using Only Two Odd Numbers
- Primitive Pythagorean triples are coprime.
- Proof of Fermat's Little Theorem
- Proof of Wilson's Theorem
- Torsion Function
- Proof of the Multiplicative Property of Totient Functions
- Euler's Totient Theorem Proof
- Euler's Totient Summation Formula Derivation
- Proof of the Chinese Remainder Theorem
- Mersenne Primes
- Sigma Functions in Number Theory
- Euclid's Derivation of the Perfect Number Formula
- Euler's Proof of the Perfect Number Theorem
- Square-and-Multiply Algorithm Proof
- Roots of Congruence Equations
- Carmichael Numbers
- Orders in Number Theory
- Proof of the Primitive Element Theorem
- Coset Decision Method
- Miller-Rabin Primality Test
- Quadratic Residues and Non-Quadratic Residues
- Proof of the Multiplicative Property of the Legendre Symbol
- Euler's Criterion
- Proof of Gauss's Law of Quadratic Reciprocity
- Primes That Leave a Remainder of 1 When Divided by 4
- Primes Divisible by 3 with a Remainder of 1: Necessary and Sufficient Conditions
- Pell's equation
- Encryption and Decryption in Cryptography
- Discrete Logarithms
- Proof of the Diffie-Hellman Key Exchange Algorithm
- ElGamal Public Key Cryptosystem Proof
- Shor's Algorithm Proof
- Smooth Primes
- Proof of Pollard's Rho Algorithm
- Discrete Logarithm Problems Solved Easily Under Certain Conditions
- Semi-prime
- Proof that the Square Root of 2 is Irrational
- Prime Factorization
- Proof of the RSA Public Key Cryptosystem
- Proof of the Goldwasser-Micali Probabilistic Key Cryptosystem
- Proof of the Pollard p-1 Factoring Algorithm
- Factorization of Semiprimes: Conditions for Easy Resolution
- Gaussian Integers
- Gaussian Ring Norm
- Proof of the Gaussian Prime Theorem
- Eisenstein Integer
- Eisenstein Ring's Norm
- Eisenstein Prime Number Theorem Proof
- List of decimals to the 10,000th
- Arithmetic Functions in Analytic Number Theory
- Arithmetic Functions' Dirichlet Convolution
- Identity for Dirichlet Products
- Inverse of Dirichlet Products
- Arithmetic Functions of Abelian groups
- Arithmetic Functions' Multiplicative Properties
- Dirichlet Product and Multiplicative Properties
- Multiplicative Function's Abelian group
- Divisor Function in Analytic Number Theory
- Analytic Number Theory: Norms
- The Moebius Function in Analytic Number Theory
- Euler's Totient Function in Analytic Number Theory
- In Analytic Number Theory
- Mobius Inversion Formula Derivation
- Analytic Number Theory and the Mangoldt Function
- Analytic Number Theory and the Liouville Function
- Arithmetic Functions' Bell Series
- Differentiation of Arithmetic Functions
- Selberg Identity Proof
- Generalized Dirichlet Product
- Generalized Dirichlet Product Representation for partial Sums of Arithmetic Functions
- P-adic Numbers in Number Theory
- Proof of Exponential Auxiliary Lemma
- Greatest Common Divisor and Coprime
- Quotient and Remainder
- Definition of Even Numbers
- Prime and Composite Numbers