Probable prime

In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific conditions. While there may be probable primes that are composite (called pseudoprimes), the condition is generally chosen in order to make such exceptions rare. Fermat's test for compositeness, which is based on Fermat's little theorem, works as follows: given an integer n, choose some integer a that is not a multiple of n; (typically, we choose a in the range 1 < a < n − 1). Calculate an − 1 modulo n. If the result is not 1, then n is composite. If the result is 1, then n is likely to be prime; n is then called a probable prime to base a. A weak probable prime to base a is an integer that is a probable prime to base a, but which is not a strong probable prime to base a (see below). For a fixed base a, it is unusual for a composite number to be a probable prime (that is, a pseudoprime) to that base. For example, up to 25 × 109, there are 11,408,012,595 odd composite numbers, but only 21,853 pseudoprimes base 2. The number of odd primes in the same interval is 1,091,987,404. Probable primality is a basis for efficient primality testing algorithms, which find application in cryptography. These algorithms are usually probabilistic in nature. The idea is that while there are composite probable primes to base a for any fixed a, we may hope there exists some fixed P

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Integers: Well Ordering and Induction

Explores well ordering, induction, Euclidean division, and prime factorization in integers.

Primality test

A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input).

Probable prime

Prime number

A prime number (or a prime) 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. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4.