RANDU is an obsolete method for generating random numbers used primarily in the 1960s and 1970s. It is a linear congruential generator (LCG) of the Park–Miller type defined by the recurrence

with the initial seed number ${\displaystyle V_{0}}$ as an odd number. It generates pseudorandom integers ${\displaystyle V_{j}}$ which are uniformly distributed in the interval [0, 2³¹ − 1], but in practical applications are often mapped into pseudorandom rationals ${\displaystyle X_{j}}$ in the interval (0, 1), by the formula

IBM's RANDU is widely considered to be one of the most ill-conceived random number generators ever designed, and was described as "truly horrible" by Donald Knuth. It fails the spectral test badly for dimensions greater than 2, as shown below.

The reason for choosing these particular values for the multiplier and modulus had been that with a 32-bit-integer word size, the arithmetic of mod 2³¹ and ${\displaystyle 65539=2^{16}+3}$ calculations could be done quickly, using bitwise operators in hardware, but the values were chosen for computational convenience, not statistical quality.

For any linear congruential generator with modulus m used to generate points in n-dimensional space, the points fall in no more than ${\displaystyle (n!\times m)^{1/n}}$ parallel hyperplanes. This indicates that low-modulus LCGs are unsuited to high-dimensional Monte Carlo simulation. For m = 2³¹ and n = 3, an LCG could have up to 2344 planes, theoretical maximum. A much tighter upper bound is proved in the same Marsaglia paper to be the sum of the absolute values of all the coefficients of the hyperplanes in standard form. That is, if the hyperplanes are of the form Ax₁ + Bx₂ + Cx₃ = some integer such as 0, 1, 2 etc, then the maximum number of planes is |A| + |B| + |C|.

Now we examine the values of multiplier 65539 and modulus 2³¹ chosen for RANDU. Consider the following calculation where every term should be taken mod 2³¹. Start by writing the recursive relation as

which after expanding the quadratic factor becomes

(because 2³² mod 2³¹ = 0) and allows us to show the correlation between three points as