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lcm(m, n) (least common multiple of m and n) is the product of all prime factors of m or n (with the largest multiplicity for m or n). gcd(m, n) × lcm(m, n) = m × n. Finding the prime factors is often harder than computing gcd and lcm using other algorithms which do not require known prime factorization.
As of 2024, it is known that F n is composite for 5 ≤ n ≤ 32, although of these, complete factorizations of F n are known only for 0 ≤ n ≤ 11, and there are no known prime factors for n = 20 and n = 24. [5] The largest Fermat number known to be composite is F 18233954, and its prime factor 7 × 2 18233956 + 1 was discovered in October 2020.
A cluster prime is a prime p such that every even natural number k ≤ p − 3 is the difference of two primes not exceeding p. 3, 5, 7, 11, 13, 17, 19, 23, ... (OEIS: A038134) All odd primes between 3 and 89, inclusive, are cluster primes. The first 10 primes that are not cluster primes are: 2, 97, 127, 149, 191, 211, 223, 227, 229, 251.
Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer n using mental or pen-and-paper arithmetic, the simplest method is trial division : checking if the number is divisible by prime numbers 2 ...
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: N = a 2 − b 2 . {\displaystyle N=a^{2}-b^{2}.} That difference is algebraically factorable as ( a + b ) ( a − b ) {\displaystyle (a+b)(a-b)} ; if neither factor equals one, it is a proper ...
A Gaussian integer is either the zero, one of the four units (±1, ±i), a Gaussian prime or composite. The article is a table of Gaussian Integers x + iy followed either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime.