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A prime sieve or prime number sieve is a fast type of algorithm for finding primes. There are many prime sieves. The simple sieve of Eratosthenes (250s BCE), the sieve of Sundaram (1934), the still faster but more complicated sieve of Atkin [1] (2003), sieve of Pritchard (1979), and various wheel sieves [2] are most common.
For example, consider the triple [20,21,29], which can be calculated from the Euclid equations with values m = 5 and n = 2. Also, arbitrarily put the coefficient of 4 in front of the x in the m term. Let m 1 = (4x + m), and let n 1 = (x + n). Hence, substituting the values of m and n:
A simple algorithm to generate a permutation of n items uniformly at random without retries, known as the Fisher–Yates shuffle, is to start with any permutation (for example, the identity permutation), and then go through the positions 0 through n − 2 (we use a convention where the first element has index 0, and the last element has index n − 1), and for each position i swap the element ...
Again, the output is half the size of the input. Beginning with a 2 b-bit input word, the top b−3 bits are used for a shift amount, which is applied to the next-most-significant 2 b−1 +2 b−3 −1 bits, and the least significant 2 b−1 bits of the result are output. The low 2 b−1 −2 b−3 −b+4 bits are discarded.
The middle n digits of the result would be the next number in the sequence and returned as the result. This process is then repeated to generate more numbers. The value of n must be even in order for the method to work – if the value of n is odd, then there will not necessarily be a uniquely defined "middle n-digits" to select from. Consider ...
If addition or subtraction is used, the maximum period is (2 k − 1) × 2 M−1. If multiplication is used, the maximum period is (2 k − 1) × 2 M−3, or 1/4 of period of the additive case. If bitwise xor is used, the maximum period is 2 k − 1. For the generator to achieve this maximum period, the polynomial: y = x k + x j + 1
The Lehmer random number generator [1] (named after D. H. Lehmer), sometimes also referred to as the Park–Miller random number generator (after Stephen K. Park and Keith W. Miller), is a type of linear congruential generator (LCG) that operates in multiplicative group of integers modulo n. The general formula is
Its base is based on prime numbers. Park-Miller generator: 1988 S. K. Park and K. W. Miller [13] A specific implementation of a Lehmer generator, widely used because it is included in C++ as the function minstd_rand0 from C++11 onwards. [14] ACORN generator: 1989 (discovered 1984) R. S. Wikramaratna [15] [16] The Additive Congruential Random ...