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 evenly by themselves and 1, and numbers that can be factored into smaller whole numbers.
What are Prime Numbers: A prime number is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself.
How to Find Prime Numbers: There are various methods to identify prime numbers such as the Sieve of Eratosthenes or the divisibility test.
Composite Numbers: A composite number is a positive integer that has at least one positive divisor other than 1 and itself.
Difference Between Prime and Composite Numbers: The main difference is that prime numbers have only two divisors, whereas composite numbers have more than two.
Patterns in Prime Numbers: Prime numbers don't follow predictable patterns or sequences, but there are some patterns, such as the Prime Number Theorem.
Factors and Multiples: Factors and multiples are related to prime numbers and composite numbers. A factor is a positive integer that divides evenly into another integer, while a multiple is the result of multiplying a number by an integer.
Prime Factorization: Prime factorization is the process of breaking down a composite number into its prime factors. It is essential in cryptography and number theory.
Common Factors and Greatest Common Factor: Common factors are divisors that two or more numbers share. The greatest common factor (GCF) is the largest common factor that two or more numbers share.
Least Common Multiple: The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more numbers.
Coprime Numbers: Coprime numbers are two or more numbers that have no common factors other than 1.
Goldbach's Conjecture: Goldbach's Conjecture states that every even number greater than 2 can be expressed as the sum of two primes.
Twin Primes: Twin primes are pairs of prime numbers that differ by two, such as 3 and 5, 5 and 7, or 11 and 13.
Mersenne Primes: Mersenne primes are prime numbers that can be expressed in the form of 2^n - 1, where n is a prime number.
Fermat's Little Theorem: Fermat's Little Theorem states that if p is a prime number and a is any positive integer not divisible by p, then a^(p-1) - 1 is divisible by p.
Euler's Totient Function: Euler's Totient Function is a mathematical function that gives the number of positive integers that are coprime to a given positive integer n.
Mersenne prime: A prime number of the form 2ⁿ − 1, where n is a positive integer.
Fermat prime: A prime number of the form 2²ⁿ + 1, where n is a positive integer.
Twin primes: Two prime numbers that differ by two.
Sophie Germain prime: A prime number p such that 2p + 1 is also prime.
Circular primes: A prime number whose digits, when rotated, generate a different prime number.
Cunningham chain: A sequence of prime numbers where each prime is of the form k×2ⁿ ± 1, where k and n are positive integers.
Abundant numbers: A number where the sum of its proper divisors is greater than the number itself.
Deficient numbers: A number where the sum of its proper divisors is less than the number itself.
Pronic numbers: A composite number that is the product of two consecutive integers.
Perfect numbers: A number where the sum of its proper divisors is equal to the number itself.
Compound numbers: A number that has more than two factors.
Hyperperfect numbers: A number where the sum of its proper divisors is one less than a power of two.
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..."