A theorem that provides a necessary and sufficient condition for a number to be prime.
Modular arithmetic: Arithmetic that is done on remainders.
Fermat's little theorem: A special case of Wilson's theorem.
Euler's totient function: Counts the number of positive integers up to a given integer that are relatively prime with that integer.
Chinese remainder theorem: A theorem that describes the solutions of a system of linear congruences with pairwise coprime moduli.
Multiplicative inverse: A number that when multiplied with another number results in 1, modularly.
Primality testing: Determining if a given integer is prime or composite.
Euler's phi function: Another term for Euler's totient function.
Modular multiplication: Multiplication that is done with remainders.
Divisibility rules: Rules that determine if a number is divisible by another number without actually dividing.
Group theory: The study of the properties of groups, which are mathematical objects that capture symmetries and transformations.
Wilson's theorem is a statement in number theory that describes a relationship between prime numbers and factorials: It states that a positive integer n is a prime number if and only if.
n-1)! ≡: Mod n).
This means that if you take the factorial of one less than a prime number and divide it by that prime number, the remainder will always be: On the other hand, if the remainder is not -1, then the number is not prime.
This theorem only works for odd prime numbers: For even prime numbers, the remainder will be 0.
Wilson's theorem is sometimes used as a primality test, although there are more efficient methods available.: Wilson's theorem states that a positive integer n is prime if and only if (n-1)! + 1 is divisible by n.
There are some variations of Wilson's theorem that involve adding or subtracting 1 from the left-hand side of the equation: These variations can have interesting properties and applications in cryptography, but they are less well-known than the original theorem.