Lehmer's method is a type of "Prime Modulus Multiplicative Linear Congruential Generator" which operates in a multiplicative group of integers modulo $m$. The genious of Lehmer's algorithm is that if the multiplier $a$ and prime modulus $m$ are properly chosen, the resulting sequence $\{x_n\}$ will be statistically indistinguishable from a sequence drawn at random from the set $\{1,2,...,m-1\}$.

The recursive formula applied in the method is:

$$x_{n+1}=a{x}_n\mathrm{mod}m.$$

There are many ways of choosing the numbers $a$, $m$ and the seed $x_0$. Different values would cause the method to reach its period faster when such numbers are not "good enough".

Let's take an example: $a=6$, $m=13$ and $x_0=1$; The resulting sequence is $\{1,6,10,8,9,2,12,7,3,5,4,11,1,...\}$, which has a period of $12$.

Choosing $a,m$ and $x_0$

In general we want our choice of $m$ to be the highest possible prime number, this comes, however, at a big computing cost. For this reason, it's very common to work with numbers $m$ that are a power of $2$. Doing so guarantees us that the period would be at most $m/4$, which is pretty low, but this period would be even smaller when $m$ is even. Due to this, a good option for $m$ is, for example $m=2^{31}-1=2147483647$.

The best choice for $x_0$ is to make it coprime with $m$, this becomes a trivial task when $m$ is prime and a bit more complicaten when it is not. Lehmer suggested $a=7^5=16807$ and $m=2^{31}-1$, which is a Mersenne prime number and $0<x_0<2^{31}-1$. This choice of $a$ generates a full period function, which means it'll have a period of $m-1$

Drawing the sequence

You can now choose different values for $a$, $m$ and $x_0$ and the drawing will paint each pixel black or white depending on weather the random number generated is smaller or greater than $0.5$. A bad choice of $a$, $m$ and $x_0$ will result in an image with a recognizable pattern.