"states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors."
Every integer greater than 1 can be written uniquely as a product of primes.
Prime numbers: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
Divisibility: The property of a number to divide another number without leaving a remainder.
Factors: Numbers that divide a particular integer evenly, leaving no remainder.
Multiples: Numbers that can be obtained by multiplying a given integer by any natural number.
Greatest Common Divisors (GCD): The largest positive integer that divides two or more integers without a remainder.
Euclidean Algorithm: A method for finding the GCD of two integers.
Prime factorization: Expressing a natural number as a product of its prime factors.
Unique factorization theorem: The statement that any positive integer can be expressed uniquely as a product of primes (up to the order in which the primes are written).
Composite numbers: Numbers that are not prime, and can be expressed as a product of two or more smaller natural numbers.
Sieve of Eratosthenes: A simple, ancient algorithm for finding all prime numbers up to a given limit.
Goldbach conjecture: The conjecture that every even integer greater than 2 can be expressed as the sum of two primes.
Prime number theorem: The distribution of prime numbers among the natural numbers.
RSA cryptography: A method of encrypting data using the properties of prime numbers.
Chinese Remainder Theorem: A theorem that describes a way to solve a system of congruences.
Bezout's Identity: A theorem that states that given any two integers, their GCD can be expressed as a linear combination of the two integers.
Modular arithmetic: A subset of arithmetic with special rules for addition, subtraction, and multiplication of integers.
Euler's Totient Function: A function that counts the number of positive integers up to a given integer that are relatively prime to it.
Farey sequences: Sequences of fractions between 0 and 1 with denominators less than a given number, used in number theory and geometry.
Fermat's Little Theorem: A theorem that states that if p is a prime number and a is any integer, then a^p: A is an integer multiple of p.
Wilson's Theorem: A theorem that states that a natural number n is prime if and only if (n-1)! + 1 is divisible by n.
Mersenne primes: Prime numbers of the form 2^n: Where n is a positive integer.
Lucas-Lehmer primality test: A primality test for Mersenne numbers that involves performing a series of modular arithmetic operations.
Perfect numbers: A number that is equal to the sum of its proper divisors.
Amicable numbers: Two numbers are amicable if each is the sum of the proper divisors of the other.
Sierpinski numbers: A sequence of integers that are conjectured to be composite, but for which no proof has been found.
Fundamental theorem of algebra: This theorem states that every non-constant polynomial with complex coefficients has at least one complex root.
Fundamental theorem of calculus: This theorem relates differentiation and integration, stating that differentiation is the inverse of integration.
Fundamental theorem of Galois theory: This theorem describes the connection between field extensions and the automorphism groups of those extensions.
Fundamental theorem of groups: This theorem states that every finite group can be decomposed into a direct product of cyclic groups.
Fundamental theorem of linear algebra: This theorem states that every vector space has a basis, and that any two bases of the same vector space have the same number of elements.
Fundamental theorem of projective geometry: This theorem characterizes projective transformations, showing that they preserve collinearity and incidence relations.
Fundamental theorem of symmetric polynomials: This theorem states that every symmetric polynomial can be expressed as a polynomial in the elementary symmetric polynomials.
"no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product."
"The requirement that the factors be prime is necessary."
"factorizations containing composite numbers may not be unique"
"This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique."
"The theorem generalizes to other algebraic structures that are called unique factorization domains..."
"...include principal ideal domains, Euclidean domains, and polynomial rings over a field."
"The theorem does not hold for algebraic integers."
"This failure of unique factorization is one of the reasons for the difficulty of the proof of Fermat's Last Theorem."
"358 years between Fermat's statement and Wiles's proof."
"The implicit use of unique factorization in rings of algebraic integers is behind the error of many of the numerous false proofs that have been written."
"also called the unique factorization theorem and prime factorization theorem."
"every integer greater than 1 can be represented uniquely as a product of prime numbers"
"up to the order of the factors"
"no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product."
"for example, 12 = 2 ⋅ 6 = 3 ⋅ 4"
"if 1 were prime, then factorization into primes would not be unique"
"This failure of unique factorization is one of the reasons for the difficulty of the proof of Fermat's Last Theorem."
"other algebraic structures that are called unique factorization domains"
"include principal ideal domains, Euclidean domains, and polynomial rings over a field."