RANDU: A Poorly Conceived
"Random" Number Generator
(IBM, SYSTEM/360)

RANDU, an LCG with m = 231, a = 216 + 3 = 65,539, c = 0 [ie., x i+1 = 65539x i mod (231)] has a major failing as per G. Marsaglia's observations regarding LCG random numbers, "Random Numbers Fall Mainly in The Planes", Proceedings of the National Academy of Sciences, 61, pp. 25-28 (1968). In particular, for an LCG with m = 231, triples produced using consecutive values fall into no more than 2,344 planes in 3-space. For RANDU, the choice of multiplier is particularly unfortunate, because RANDU triples fall in just 15 planes. This realization was a motivator for the development of the spectral test for n-space uniformity. This matters because many problems addressed by the Monte Carlo method (eg., the determination of random neutron diffusion in fissile material) require large numbers of random points in n-space. The point plot below has 32,000 "random" triples generated using RANDU and demonstrates that they distribute in just 15 planes; ie., RANDU fails badly for generating random n-tuples with n = 3. The plot image was generated using an application written by Y. S. Chua, UNF [2004] 96,000 consecutive RANDU values were collapsed to points of the form (x 0, x 1, x 2), (x 3, x 4, x 5), . . . with the number stream ranged to 0 - 1000 by using FLOOR(0.5+1000x i / 231) Note that for RANDU, a triple of successive values (x i, x i+1, x i+2) has the property that 9x i - 6x i+1 + x i+2 = 0 (mod 231) [9x i - 6x i+1 + x i+2 = 9x i - 6ax i + a2x i = (a - 3)2 x i = (216)2 x i= 231 2x i = 0 (mod 231)] Hence Marsaglia's Theorem applies, so each of the triples lies on one of the set of parallel planes defined by the equations 9x - 6y + z = 0, ±1, ±2, .... with at most 9+6+0=15 of these intersecting the 231 x 231 x 231 cube containing the triples.       animated view