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This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.
After 1 and 9, 225 is the third smallest number n for which σ(φ(n)) = φ(σ(n)), where σ is the sum of divisors function and φ is Euler's totient function. [8] 225 is a refactorable number. [9] 225 is the smallest square number to have one of every digit in some number base (225 is 3201 in base 4) [10] 225 is the first odd number with ...
Ω(n), the prime omega function, is the number of prime factors of n counted with multiplicity (so it is the sum of all prime factor multiplicities). A prime number has Ω(n) = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 (sequence A000040 in the OEIS). There are many special types of prime numbers. A composite number has Ω(n) > 1.
The progressions of numbers that are 0, 3, or 6 mod 9 contain at most one prime number (the number 3); the remaining progressions of numbers that are 2, 4, 5, 7, and 8 mod 9 have infinitely many prime numbers, with similar numbers of primes in each progression.
These numbers have been proved prime by computer with a primality test for their form, ... 225 9693×2 7767343 − 1 17 November 2023 2,338,208 226 5×2 7755002 − 1
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1. [1] [2] The numbers p corresponding to Mersenne primes must themselves be prime, although the vast majority of primes p do not lead to Mersenne primes—for example, 2 11 − 1 = 2047 = 23 × 89. [3]
It is the fiftieth prime number, and a regular prime. [1] It is also a full reptend prime, meaning that the decimal expansion of the unit fraction 1/229 repeats periodically with as long a period as possible. [2] With 227 it is the larger of a pair of twin primes, [3] and it is also the start of a sequence of three consecutive squarefree ...