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Certain number-theoretic methods exist for testing whether a number is prime, such as the Lucas test and Proth's test. These tests typically require factorization of n + 1, n − 1, or a similar quantity, which means that they are not useful for general-purpose primality testing, but they are often quite powerful when the tested number n is ...
The most popular Java Edition server is Hypixel, which, released in April 2013, has had over 20 million unique players. [3] [4] In 2021, CubeCraft Games, released in December 2012 on Java Edition and in 2018 on Bedrock Edition, [20] had over 30 million unique server connections, and a peak player count of more than 57,000 concurrent players. [21]
Mineplex was a Minecraft minigame server created in 2013 by Gregory Bylos and Jarred van de Voort. [4] [5] In 2016, Mineplex had millions of unique players monthly. [6]At its peak, the server had around 20,000 concurrent players at any given time. [7]
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 first working version that was widely deployed was assigned version number 4. [10] A separate protocol based on reliable connections was developed and assigned version 5. IP version 7 was chosen in 1988 by R. Ullmann as the next IP version because he incorrectly assumed that version 6 was in use for ST-II.
For these numbers, repeated application of the Fermat primality test performs the same as a simple random search for factors. While Carmichael numbers are substantially rarer than prime numbers (Erdös' upper bound for the number of Carmichael numbers [ 3 ] is lower than the prime number function n/log(n) ) there are enough of them that Fermat ...
Primality certificates allow the primality of a number to be rapidly checked without having to run an expensive or unreliable primality test. "Succinct" usually means that the proof should be at most polynomially larger than the number of digits in the number itself (for example, if the number has b bits, the proof might contain roughly b 2 bits).
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 it was conjectured to contain all odd primes, even though it is rather inefficient.