Quote: "A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers."
Numbers that can only be divided by 1 and itself without leaving a remainder.
Divisibility: In number theory, divisibility refers to the property of a number that can be divided evenly by another number without leaving a remainder.
Factorization: Factorization is a technique used to list out all the prime factors of a given number.
Sieve of Eratosthenes: It is a method used to generate all prime numbers up to a given limit.
Euclidean algorithm: It is a method used to find the greatest common divisor of two given numbers.
Coprime numbers: Coprime numbers are two numbers which have no common factors other than 1.
Euler's Totient Function: It is a function that calculates the number of positive integers less than or equal to a given number n which are coprime to n.
Fermat's Little Theorem: It is a theorem that states that for any prime number p and any integer a not divisible by p, a^(p-1) ≡ 1(mod p).
Wilson's Theorem: It is a theorem that states that if p is a prime number, then (p-1)! ≡ -1(mod p).
Goldbach's Conjecture: It is an unproven conjecture that states that every even integer greater than 2 can be expressed as the sum of two prime numbers.
Primality Testing Algorithms: It is a set of algorithms used to determine whether a given number is prime or composite.
Twin Prime Conjecture: It is an unproven conjecture that states that there are infinitely many twin prime numbers.
Mersenne Prime: A Mersenne prime is a prime number that is one less than a power of two.
Prime Decomposition: Prime decomposition is the process of representing a given non-zero positive integer as a product of its prime factors.
Prime gap: The prime gap is the difference between two consecutive prime numbers.
Prime number theorem: The Prime Number Theorem is an asymptotic result that says, as n approaches infinity, the number of primes less than or equal to n is approximately n/log n.
Mersenne primes: These are prime numbers that take the form 2^n -1, where n is a positive integer.
Fermat primes: These are prime numbers that take the form 2^(2^n) +1, where n is a positive integer.
Sophie Germain primes: These are prime numbers p such that 2p+1 is also a prime number.
Wilson primes: These are primes number p such that (p-1)! + 1 is divisible by p^2.
Proth prime: These are prime numbers of the form k*2^n+1, where k is an odd number and n is a positive integer.
Quote: "A natural number greater than 1 that is not prime is called a composite number."
Quote: "5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself."
Quote: "4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4."
Quote: "Every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order."
Quote: "The property of being prime is called primality."
Quote: "Trial division tests whether n is a multiple of any integer between 2 and √n."
Quote: "Faster algorithms include the Miller–Rabin primality test."
Quote: "As of December 2018, the largest known prime number is a Mersenne prime with 24,862,048 decimal digits."
Quote: "The prime number theorem... says that the probability of a randomly chosen large number being prime is inversely proportional to its number of digits, that is, to its logarithm."
Quote: "These include Goldbach's conjecture... and the twin prime conjecture."
Quote: "Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors."
Quote: "Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers."
Quote: "Objects that behave in a generalized way like prime numbers include prime elements and prime ideals."
Quote: "As demonstrated by Euclid around 300 BC, there are infinitely many primes."
Quote: "No known simple formula separates prime numbers from composite numbers."
Quote: "The distribution of primes within the natural numbers in the large can be statistically modeled."
Quote: "The AKS primality test always produces the correct answer in polynomial time but is too slow to be practical."
Quote: "Particularly fast methods are available for numbers of special forms, such as Mersenne numbers."
Quote: "For example, 5 is prime..."