Prime Numbers and Primality Testing

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.
A natural number greater than 1 that is not prime is called a composite number.

A prime number can only be divided by 1 and itself.
For example, 5 is prime because the only ways of writing it as a product are 1 × 5 or 5 × 1.
The property of being prime is called primality.

A simple but slow method of checking primality is trial division, which tests whether a number is a multiple of any integer between 2 and its square root.
Faster algorithms include the Miller–Rabin primality test and the AKS primality test.

There are infinitely many primes, as demonstrated by Euclid around 300 BC.
No known simple formula separates prime numbers from composite numbers.
The distribution of primes within the natural numbers can be statistically modelled.

The prime number theorem states that the probability of a randomly chosen large number being prime is inversely proportional to its number of digits, or its logarithm.

Prime numbers are used in public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors.
In abstract algebra, objects that behave like prime numbers include prime elements and prime ideals.

Goldbach's conjecture states that every even integer greater than 2 can be expressed as the sum of two primes.
The twin prime conjecture states that there are infinitely many pairs of primes that differ by two.
These questions have spurred the development of various branches of number theory.

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.
A natural number greater than 1 that is not prime is called a composite number.

A prime number can only be divided by 1 and itself.
For example, 5 is prime because the only ways of writing it as a product are 1 × 5 or 5 × 1.
The property of being prime is called primality.

A simple but slow method of checking primality is trial division, which tests whether a number is a multiple of any integer between 2 and its square root.
Faster algorithms include the Miller–Rabin primality test and the AKS primality test.

There are infinitely many primes, as demonstrated by Euclid around 300 BC.
No known simple formula separates prime numbers from composite numbers.
The distribution of primes within the natural numbers can be statistically modelled.

The prime number theorem states that the probability of a randomly chosen large number being prime is inversely proportional to its number of digits, or its logarithm.

Prime numbers are used in public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors.
In abstract algebra, objects that behave like prime numbers include prime elements and prime ideals.

Goldbach's conjecture states that every even integer greater than 2 can be expressed as the sum of two primes.
The twin prime conjecture states that there are infinitely many pairs of primes that differ by two.
These questions have spurred the development of various branches of number theory.

A prime number is a positive integer that is divisible only by itself and 1.
Prime numbers are greater than 1 and have exactly two distinct positive divisors: 1 and themselves.

Prime numbers are always odd, except for the number 2, which is the only even prime number.

Prime numbers play a fundamental role in number theory and are used in many mathematical concepts, such as cryptography, algebra, and geometry.

A prime number is a positive integer divisible only by itself and 1.
It is a natural number greater than 1 with no positive divisors other than 1 and itself.
The smallest prime number is 2.

Trial Division is a method to check if a number is prime by dividing it by all prime numbers less than or equal to its square root.
Fermat's Little Theorem states that if p is prime, then a^p ≡ a (mod p) for any integer a.
Miller-Rabin Primality Test is a probabilistic algorithm to determine if a number is prime or composite.

Euclid's Theorem states that there are infinitely many prime numbers.
The Prime Number Theorem approximates the distribution of prime numbers using the logarithmic integral function.

Prime Factorization is the process of expressing a composite number as a product of prime numbers.
The Fundamental Theorem of Arithmetic states that every positive integer can be expressed as a unique product of prime numbers, except for the order in which they are listed.
Factor Trees are a graphical method to find prime factors.
The Sieve of Eratosthenes is an algorithm to find all prime numbers up to a given number.
Pollard's Rho Algorithm is a method to find prime factors of large numbers.