Linear Congruential Random Number Generator (LCG)#

Stage 4 · 🧩 Sampling, Partitioning & Segmentation · Lesson 24 of 56 · intermediate

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Important

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Randomness you can repeat#

The sampling of the last lesson needs a source of randomness — but for science it must also be reproducible, so that a colleague running the same code gets the same split. Computers square this circle with pseudo-random number generators: deterministic recipes that produce sequences which look random. The linear congruential generator (LCG) is the classic, and the simplest to understand.

The recurrence#

An LCG produces each number from the previous one by a single formula:

\[X_{n+1} = (a\,X_n + c) \bmod m,\]

where \(m\) is the modulus, \(a\) the multiplier, \(c\) the increment, and \(X_0\) the starting value. Each \(X_n\) is an integer in \([0, m)\); dividing by \(m\) rescales it to a fraction in \([0, 1)\). From three constants and a start value, an endless stream of “random” numbers follows.

The seed and the period#

The starting value \(X_0\) is the seed. Because the formula is deterministic, the same seed always yields the same sequence — precisely what makes results reproducible (random.seed / numpy.random.seed set it). Being finite, the sequence must eventually repeat; the length before it does is the period, at most \(m\). Careful choices of \(a\), \(c\) and \(m\) (the Hull–Dobell conditions) achieve the full period.

Simple, but dated#

The LCG is prized for being fast, tiny and easy to reason about, and it makes seeded reproducibility concrete. But its statistical quality is limited — successive values fall on detectable lattice patterns, and the low-order bits are weakly random. Modern libraries therefore default to stronger generators (numpy now uses a PCG64 generator by default), while keeping the same crucial habit: set a seed so your sampling, splitting and modelling can be reproduced exactly.

Hint