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The first 1000 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms. 1 is neither prime nor composite. The first 1000 prime numbers
Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2 p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1.
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.
They are named after Pierre de Fermat, who conjectured that all such numbers are prime. The first five of these numbers – 3, 5, 17, 257, and 65,537 – are prime, [172] but is composite and so are all other Fermat numbers that have been verified as of 2017. [173]
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 a n = 3 + 4 n {\displaystyle a_{n}=3+4n} for 0 ≤ n ≤ 2 {\displaystyle 0\leq n\leq 2} .
The first number after 1 for wheel 2 is 5; note it as a prime. Now form wheel 3 with length 5 × 6 = 30 by first extending wheel 2 up to 30 and then deleting 5 times each number in wheel 2 (in reverse order!), to get 1 2 3 5 7 11 13 17 19 23 25 29. The first number after 1 for wheel 3 is 7; note it as a prime.
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The largest known prime number is 2 136,279,841 − 1, a number which has 41,024,320 digits when written in the decimal system. It was found on October 12, 2024, on a cloud-based virtual machine volunteered by Luke Durant, a 36-year-old researcher from San Jose, California, to the Great Internet Mersenne Prime Search (GIMPS).