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# Long Range Correlations in Linear Congruential Generators

Long Range Correlations in Linear Congruential Generators
O E Percus
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. . , x^_ ^ ) is a permutation of {0, 1, 2, m-1) . For a fixed k, we now ask whether there exists a guaranteed linear relation among entries in the chain separated by 2 ; S c. X - (mod 2^) . (3) J n+j2'^ Substituting (2) into (3), this will be the case for given x^ and b if f(a)xQ - (mod 2^) .
^^^^^ b - (mod 2^ . (4) b k where f(0) - E c. A-^ J The case b - 0, Xq odd, was studied by Filk, Marcu, and Fredenhagen, here one only requires f(a) Â« (mod 1^-), (5) independently of xâ€ž . The basic relati
...on of type (5) is the Euler totient formula' which becomes in the present case a -1=0 (mod 2^) . (6) However, there are numerous polynomials of lower degree that satisfy (5). For relations between {x ) separated by 2, these are most readily generated from the primitive case &- - 1 - (mod 2"^^^^) (7) [31 for suitable 7(k) . The totient formula guarantees that 7(k) > k+1, but the decomposition Â«k k-1 â€žj a - 1 - (a-1) n (a + 1) (8) j-0 establishes at once the result that if a â–  1 (mod 2 ), then 7(k) > k+a.

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