RANDU

Three-dimensional plot of 100,000 values generated with RANDU. Each point represents 3 consecutive pseudorandom values. It is clearly seen that the points fall in 15 two-dimensional planes.

RANDU is a linear congruential pseudorandom number generator of the Park–Miller type, which has been used since the 1960s. It is defined by the recurrence:

$V_{j+1}=65539\cdot V_{j}\,{\bmod {\,}}2^{31}\,$

with $\scriptstyle V_{0}$ odd. It generates pseudorandom integers $\scriptstyle V_{j}$ in the interval [1, 2³¹ − 1] that in practical applications are often mapped into pseudorandom rationals $\scriptstyle X_{j}$ in the interval (0, 1) by the formula:

$X_{j}={\frac {V_{j}}{2^{31}}}$.

It is widely considered to be one of the most ill-conceived random number generators designed. Notably, it fails the spectral test badly for dimensions greater than 2, and every result is odd.

The reason for choosing these particular values is that with a 32 bit integer word size the arithmetic of mod 2³¹ and $65539=2^{16}+3$ calculations could be done quickly. To show the problem with these values consider the following calculation where every term should be taken mod 2³¹, we start by writing the recursive relation as:

$x_{k+2}=(2^{16}+3)x_{k+1}=(2^{16}+3)^{2}x_{k}\,$

which becomes, after expanding the quadratic factor:

$x_{k+2}=(2^{32}+6\cdot 2^{16}+9)x_{k}=[6\cdot (2^{16}+3)-9]x_{k}\,$

because 2³² mod 2³¹ = 0

and allows us to show the enormous correlation between three points as:

$x_{k+2}=6x_{k+1}-9x_{k}\,$

As a result of this correlation, the points in three dimensional space (mod 2³¹) fall in a comparatively small number of planes, 15 to be exact. As a result of the wide use of RANDU in the early 1970s, many results from that time are seen as suspicious.

This misbehavior was already detected in 1963 on a 36-bit computer, and carefully reimplemented on the 32-bit IBM360.

Sample output

The start and end of the RANDU’s output period for the initial seed $\scriptstyle V_{0}=1$ is:

1, 65539, 393225, 1769499, 7077969, 26542323, …, 2141591611, 388843697, 238606867, 79531577, 477211307, 1 (sequence A096555 in the OEIS)

Quotations

…its very name RANDU is enough to bring dismay into the eyes and stomachs of many computer scientists!

— Donald Knuth

One of us recalls producing a "random" plot with only 11 planes, and being told by his computer center's programming consultant that he had misused the random number generator: "We guarantee that each number is random individually, but we don't guarantee that more than one of them is random." Figure that out.

— W. H. Press et al.