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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
Generator Date First proponents References Notes Middle-square method: 1946 J. von Neumann [1] In its original form, it is of poor quality and of historical interest only. Lehmer generator: 1951 D. H. Lehmer [2] One of the very earliest and most influential designs. Linear congruential generator (LCG) 1958 W. E. Thomson; A. Rotenberg [3] [4]
The second row is the same generator with a seed of 3, which produces a cycle of length 2. Using a = 4 and c = 1 (bottom row) gives a cycle length of 9 with any seed in [0, 8]. A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation.
In September 1949, he presented the pseudorandom number generator now known as the Lehmer random number generator. [4] D. H. Lehmer wrote the article "The Machine Tools of Combinatorics," which is the first chapter in Edwin Beckenbach's Applied Combinatorial Mathematics (1964). [5] It describes methods for producing permutations, combinations, etc.
In computational number theory, Marsaglia's theorem connects modular arithmetic and analytic geometry to describe the flaws with the pseudorandom numbers resulting from a linear congruential generator. As a direct consequence, it is now widely considered that linear congruential generators are weak for the purpose of generating random numbers.
Couture and L'Ecuyer [3] have proved the surprising result that the lattice associated with a multiply-with-carry generator is very close to the lattice associated with the Lehmer generator it simulates. Thus, the mathematical techniques developed for Lehmer generators (such as the spectral test) can be applied to multiply-with-carry generators.
Lehmer sieves are mechanical devices that implement sieves in number theory. Lehmer sieves are named for Derrick Norman Lehmer and his son Derrick Henry Lehmer . The father was a professor of mathematics at the University of California, Berkeley at the time, and his son followed in his footsteps as a number theorist and professor at Berkeley.
The Lehmer generator, designed by Derrick Henry Lehmer in the 1940s, has been foundational in the development of pseudo-random number generation. Its simplicity and efficiency in generating sequences make it an attractive choice, especially in fields like simulations, statistical sampling, and cryptography.