Notes

The linear congruential generator

JAEV Notes · October 2026 · 3 min read

Left: a circle with 128 tick marks, criss-crossed by red straight lines that join the numbers of a sequence one after the other; only every fourth tick is long. Right: a square in which 300 black dots lie on five parallel slanting lines. Small grey labels under each.
Diagram by JAEV: Georg Nees's rule “multiply by 5, keep the remainder”. Left: with remainder after 128, starting from 1, the red lines join each number to the next; after 32 numbers the sequence is back at 1. Right: with Nees's full 2³¹, each dot is a number and the next one (300 pairs, JAEV's seed); all fall on five lines. CC BY 4.0.

A linear congruential generator is a simple, long-used way for a computer to produce numbers that look random. It takes a number, multiplies it by a constant and adds another constant, then divides by a third and keeps only the remainder. That remainder is the next number, and the rule is applied to it again.1 The mathematician D. H. Lehmer described the form without addition in a paper given in 1949 and printed in 1951; W. E. Thomson and A. Rotenberg published the form with addition in 1958.1, 2

How it works

Written as a formula: next = (a × x + c) mod m, where the first x is the seed. Every number lies between 0 and m − 1. Dividing by m turns it into a fraction between 0 and 1, which a program can scale to any range it needs.1

The computer-art pioneer Georg Nees used the simplest version. His dissertation calls the method "extremely simple": multiply by 5 and keep the remainder after dividing by 2³¹.3 The same generator decided the shifts and turns of the squares in his Schotter.3 To explain it, Nees used a small version that keeps the remainder after 128, which repeats after 32 numbers.3 Started from 1, it gives 1, 5, 25, 125, 113, 53 and so on; the left half of the picture above draws that loop.

Why it repeats

There are only m possible remainders, so sooner or later a number comes back and the sequence repeats. With an added constant and well-chosen values, a generator can run through all m numbers before that happens.1 Nees called a generator with a short period "almost always unsuitable" for his work; the period, he wrote, must be so long that "we do not notice it".3

Sketch by JAEV: a small generator, next = a × x mod m, with m 128, 256 or 512 and a random multiplier and seed. Left: the numbers as a clock, each line joining a number to the next until the sequence returns to its seed. Right: each dot is a number and the one after it; the dots line up instead of filling the square evenly. Code CC BY 4.0.

The pattern inside

The numbers are not independent. If each number and the next are drawn as a point, the points fall on a limited number of straight lines, and in three dimensions on planes.1 With the multiplier 5 the reason is plain: the next number is 5x minus 0, 1, 2, 3 or 4 times the modulus, so every pair lies on one of five lines (right half of the picture).

The structure is always there: for any constants, the points lie on at most a certain number of planes, so choosing the constants is a matter of making that number large.1 These generators must not be used where secrecy matters, as in cryptography.1

Still in use

Java's standard class java.util.Random uses "a linear congruential formula" on a 48-bit number.4 Processing, the language many generative artists learned on, gets its random() from that class, so a sketch that calls randomSeed() is in effect setting the seed of a linear congruential generator.5

JAEV's exhibition Content takes its chance from a hash instead: it turns a text into a long number, so that "the same text always gives the same work"; see the note on SHA-256 as a source of chance.6

Sources

  1. “Linear congruential generator”. Wikipedia
    The recurrence with multiplier a, increment c and modulus m; “The Lehmer generator was published in 1951”, the LCG “in 1958 by W. E. Thomson and A. Rotenberg”; Hull–Dobell theorem; points on hyperplanes; low bits; “must not be used, for cryptographic applications”.
  2. “Lehmer random number generator”. Wikipedia
    X(k+1) = a · X(k) mod m; Lehmer, “Mathematical methods in large-scale computing units” (1949, published 1951); power-of-two modulus: “the period is at most m/4”, seed odd, a ≡ ±3 (mod 8); Park and Miller 1988, m = 2³¹ − 1, a = 16807, “Random Number Generators: Good Ones Are Hard To Find”; 48271 in 1993.
  3. Zellyn Hunter, “Schotter – Georg Nees – Part 2”, 2024
    Nees: “The method we will use to generate random numbers is extremely simple”; multiply by 5, remainder modulo 2³¹; toy version modulo 128 with a 32-number period; “random generators with such a short period as the algorithm presented here are almost always unsuitable for our investigations”; the period must be long enough that “we do not notice it”.
  4. “Class Random”. Java Platform SE 8 API documentation, Oracle
    “The class uses a 48-bit seed, which is modified using a linear congruential formula.” Multiplier 0x5DEECE66D, addend 0xB.
  5. PApplet.java, Processing 4 source code. GitHub
    random() calls internalRandom.nextFloat(), where internalRandom = new Random() (java.util.Random); randomSeed() calls internalRandom.setSeed(seed).
  6. “Content”, JAEV. JAEV, Lund
    “The text is turned into a fingerprint, a long number that changes completely if a single character changes. … The same text always gives the same work.”

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JAEV Notes, October 2026.
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JAEV Notes, “The linear congruential generator”, jaev.gallery/notes/linear-congruential-generator/, 2026.
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