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All prime numbers from 31 to 6,469,693,189 for free download. Lists of Primes at the Prime Pages. The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random prime 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". MathWorld.
These numbers have been proved prime by computer with a primality test for their form, for example the Lucas–Lehmer primality test for Mersenne numbers. Φ 3 ( x ) {\displaystyle \Phi _{3}(x)} is the third cyclotomic polynomial , defined as x 2 + x + 1 {\displaystyle x^{2}+x+1} .
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 not all primes p lead to Mersenne primes—for example, 2 11 − 1 = 2047 = 23 × 89 . [ 3 ]
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