Marsaglia's theorem

Consider a Lehmer random number generator with

${\displaystyle r_{i+1}\equiv kr_{i}\mod m}$

for any modulus ${\displaystyle m}$ and multiplier ${\displaystyle k}$ where each ${\displaystyle 0<r_{i}<m}$, and define a sequence

${\displaystyle u_{1}={\frac {r_{1}}{m}},u_{2}={\frac {r_{2}}{m}},u_{3}={\frac {r_{3}}{m}},\ldots }$

Define the points

${\displaystyle \pi _{1}=(u_{1},\ldots ,u_{n}),\pi _{2}=(u_{2},\ldots ,u_{n+1}),\pi _{3}=(u_{3},\ldots ,u_{n+2}),\ldots }$

on a unit ${\displaystyle n}$-cube formed from successive terms of the sequence of ${\displaystyle u}$. With such a multiplicative number generator, all ${\displaystyle n}$-tuples of resulting random numbers lie in at most ${\displaystyle (n!m)^{1/n}}$ hyperplanes. Additionally, for a choice of constants ${\displaystyle c_{1},c_{2},\ldots ,c_{n}}$ which satisfy the congruence

${\displaystyle c_{1}+c_{2}k+c_{3}k^{2}+\cdots +c_{n}k^{n-1}\equiv 0\mod m,}$

there are at most ${\displaystyle |c_{1}|+|c_{2}|+\cdots +|c_{n}|}$ parallel hyperplanes which contain all ${\displaystyle n}$-tuples produced by the generator. Proofs for these claims may be found in Marsaglia's original paper. [2]