Marsaglia's theorem

In computational number theory, Marsaglia's theorem connects modular arithmetic and analytic geometry to describe the flaws with the pseudorandom numbers resulting from a linear congruential generator.

As a direct consequence, it is now widely considered that linear congruential generators are weak for the purpose of generating random numbers.

Particularly, it is inadvisable to use them for simulations with the Monte Carlo method or in cryptographic settings, such as issuing a public key certificate, unless specific numerical requirements are satisfied.

Poorly chosen values for the modulus and multiplier in a Lehmer random number generator will lead to a short period for the sequence of random numbers.

Marsaglia's result may be further extended to a mixed linear congruential generator.

, and in three dimensions, it shows that all the points fall into at most

The actual RANDU algorithm, which uses

All the points in fact fall into 15 planes.

Consider a Lehmer random number generator with for any modulus

, and define a sequence Define the points on a unit

-cube formed from successive terms of the sequence of

With such a multiplicative number generator, all

-tuples of resulting random numbers lie in at most

Additionally, for a choice of constants

which satisfy the congruence there are at most

parallel hyperplanes which contain all

-tuples produced by the generator.

Proofs for these claims may be found in Marsaglia's original paper.