Glossary
What is a linear congruential generator?
A linear congruential generator (LCG) is the classic pseudorandom number generator: pick a modulus m, a multiplier a and an increment c, and compute each number from the last as X(n+1) = (a X(n) + c) mod m. It is fast and tiny, and with good parameters it is fine for many simulations.
Choosing the parameters
The sequence must eventually repeat, at most every m steps. Knuth gives exact conditions for the full period m when c is not zero: c and m share no factor, a - 1 is divisible by every prime factor of m, and by 4 if m is. He also defines potency, which rules out multipliers that pass that test but produce obviously patterned output.
How to tell if it is any good
Full period is necessary, not sufficient. The rest of Volume 2's chapter 3 is about testing: empirical tests like the chi-square and gap tests, and the spectral test, which measures how the points (X(n), X(n+1), ...) fall on a few parallel planes. An LCG with a bad multiplier passes casual checks and fails the spectral test badly. Start at §3.2.1.
Questions
Should I use an LCG today?
For reproducible simulations or games it can be fine with well-tested parameters. Never for anything security related, and modern generators are usually a better default.