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Integers R n that are the smallest to give at least n primes from x/2 to x for all x ... All prime numbers from 31 to 6,469,693,189 for free download.
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.
The smallest composite Mersenne number with prime exponent n is 2 11 − 1 = 2047 = 23 × 89. ... 2008. This was the eighth Mersenne prime discovered at UCLA.
By 1772, Leonhard Euler had proven that 2,147,483,647 is a prime. The number 2147483647 is the eighth Mersenne prime, equal to 2 31 − 1. It is one of only four known double Mersenne primes. [1] The primality of this number was proven by Leonhard Euler, who reported the proof in a letter to Daniel Bernoulli written in 1772. [2]
19, alongside 109, 1009, and 10009, are all prime (with 109 also full reptend), and form part of a sequence of numbers where inserting a digit inside the previous term produces the next smallest prime possible, up to scale, with the composite number 9 as root. [17] 100019 is the next such smallest prime number, by the insertion of a 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.
73 is the 21st prime number, and emirp with 37, the 12th prime number. [1] It is also the eighth twin prime, ... and the smallest prime congruent to 1 modulo 24, ...
31 is the 11th prime number. It is a superprime and a self prime (after 3, 5, and 7), as no integer added up to its base 10 digits results in 31. [1] It is the third Mersenne prime of the form 2 n − 1, [2] and the eighth Mersenne prime exponent, [3] in-turn yielding the maximum positive value for a 32-bit signed binary integer in computing: 2,147,483,647.