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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.
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.
All instances of log (x) without a subscript base should be interpreted as a natural logarithm, also commonly written as ln (x) or loge(x). In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they ...
Rowland (2008) proved that this sequence contains only ones and prime numbers. However, it does not contain all the prime numbers, since the terms gcd(n + 1, a n) are always odd and so never equal to 2. 587 is the smallest prime (other than 2) not appearing in the first 10,000 outcomes that are different from 1. Nevertheless, in the same paper ...
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 ...
The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents p. As of 2023, there are 51 known Mersenne primes (and therefore perfect numbers), the largest 17 of which have been discovered by the distributed computing project Great Internet Mersenne Prime Search, or GIMPS. [2]
In number theory, primes in arithmetic progressionare any sequenceof at least three prime numbersthat are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by an=3+4n{\displaystyle a_{n}=3+4n}for 0≤n≤2{\displaystyle 0\leq n\leq 2}. According to the Green–Tao theorem, there ...
There are 808,675,888,577,436 twin prime pairs below 10 18. [20] [21] An empirical analysis of all prime pairs up to 4.35 × 10 15 shows that if the number of such pairs less than x is f (x) ·x /(log x) 2 then f (x) is about 1.7 for small x and decreases towards about 1.3 as x tends to infinity.