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The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range. Interface to a list of the first 98 million primes (primes less than 2,000,000,000) Weisstein, Eric W. "Prime Number Sequences" .
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.
For example, 3 is a Mersenne prime as it is a prime number and is expressible as 22 − 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, 211 − 1 = 2047 = 23 × 89. [3] Meanwhile, perfect numbers are natural numbers that equal ...
Therefore, every prime number other than 2 is an odd number, and is called an odd prime. [10] Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in ...
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 ...
Mersenne primes (of form 2^ p − 1 where p is a prime) In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century.
Primes in arithmetic progression. Set of prime numbers linked by a linear relationship. In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by for .
An odd prime number p is defined to be regular if it does not divide the class number of the p th cyclotomic field Q (ζp), where ζp is a primitive p th root of unity. The prime number 2 is often considered regular as well. The class number of the cyclotomic field is the number of ideals of the ring of integers Z (ζp) up to equivalence.