RANDU

RANDU^[1] is a linear congruential pseudorandom number generator (LCG) of the Park–Miller type, which was used primarily in the 1960s and 1970s.^[2] It is defined by the recurrence

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

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 [1, 2³¹ − 1], but in practical applications are often mapped into pseudorandom rationals ${\displaystyle X_{j}}$ in the interval (0, 1), by the formula

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

IBM's RANDU is widely considered to be one of the most ill-conceived random number generators ever designed,^[3] and was described as "truly horrible" by Donald Knuth.^[4] 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.