Here is a list of some important mathematical formulas and identities that are commonly used in number theory:
- Euclid’s formula: a formula for generating Pythagorean triples (a, b, c) such that a^2 + b^2 = c^2. It states that: a = m^2 – n^2, b = 2mn, c = m^2 + n^2, where m and n are positive integers and m > n.
- Prime factorization theorem: a result that states that every positive integer greater than 1 can be written as a unique product of prime numbers.
- The binomial theorem: a result that expresses the power (x + y)^n as a sum of binomial coefficients.
- Euler’s formula: an important formula in number theory that relates the sum of the roots of an equation to the coefficients of the equation.
- Euler’s Totient function: an important function in number theory that gives the count of integers less than a given integer n that are relatively prime to n. This function is represented by the Greek symbol ϕ(n)
- Gauss’s Law of Quadratic Reciprocity: a fundamental theorem of number theory that gives a criterion for the solvability of certain types of Diophantine equations.
- Jacobi’s symbol: a generalization of the Legendre symbol that gives a criterion for determining the solvability of certain types of Diophantine equations.
- Fermat’s Little Theorem: a result that states that if p is a prime number and a is an integer not divisible by p, then a^p-a is divisible by p.
- Euler’s Criterion: a result that states that if p is an odd prime and a is an integer not divisible by p, then a^((p-1)/2) ≡ (a/p) (mod p),
- Euclidean algorithm: a method for finding the greatest common divisor (GCD) of two integers.
- Chinese Remainder theorem: a result that states that if two integers a and b are relatively prime, then the system of congruences x ≡ a (mod m) and x ≡ b (mod n) has a unique solution.
- Euler’s totient function: a function that counts the number of positive integers up to a given integer n that are relatively prime to n.
- Wilson’s theorem: a result that states that if p is a prime number, then (p-1)! + 1 is divisible by p.
- Quadratic Reciprocity: a result that gives a criterion for determining the solvability of a particular type of Diophantine equation.
- Euler product formula: a formula expressing the Riemann zeta function as an infinite product of simpler functions.
- Möbius inversion formula: an identity that relates the values of a function to the values of its inverse.
- Ramanujan-Nagell theorem: a theorem that gives criterion for an integer to be the sum of two squares.
These formulas and identities play a crucial role in many areas of number theory, including cryptography, coding theory, and Diophantine equations.
What is Number Theory?
Number theory is a branch of mathematics that deals with the properties of integers and integer-valued functions. It is a very old area of mathematics, with roots going back to ancient Greece and the work of Pythagoras, Euclid, and others. Number theorists study properties of the integers and related objects like prime numbers, Diophantine equations, and modular forms.
Some of the fundamental questions that number theory addresses include:
- Prime numbers: The distribution and behavior of prime numbers and their role in other areas of mathematics.
- Diophantine equations: The solutions of equations where the unknowns are integers and the solutions are also integers.
- Modular forms: The study of complex-valued functions that are periodic with respect to a lattice, which are important in many areas of mathematics and physics.
- Algebraic number theory: The study of algebraic integers and their properties.
- Arithmetic geometry: The study of the solutions of equations over the integers, in the context of algebraic geometry.
- Analytic number theory: The study of the distribution of prime numbers and other number-theoretic functions using tools from complex analysis and Fourier analysis.
Number theory has many applications in other areas of mathematics, as well as in computer science, cryptography, physics, and other areas. Its basic principles, techniques and findings are used in many mathematical and non-mathematical subjects.