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  "body": "\n## Definition\n\nLet $m \\ge 2$ be an integer, the modulus, and let $a$, $c$ and $X_0$ be integers with $0 \u003c a \u003c m$, $0 \\le c \u003c m$ and $0 \\le X_0 \u003c m$.\nThe *linear congruential generator* with multiplier $a$, increment $c$ and seed $X_0$ is a [[pseudorandom number generator|C5FG6ARWVV]].\nIts state space is $\\set{0, \\ldots, m-1}$ and its states are\n$$X_{n+1} = (a X_n + c) \\bmod m, \\qquad n \\ge 0,$$\nwhere $\\bmod$ is the [[modulo operation|03BR7SAX0Q]].\nIts outputs are the numbers $u_n = X_n / m \\in [0, 1)$.\nIf $c = 0$ it is called a *multiplicative* congruential generator, or Lehmer generator.\n\n## Remarks\n\nExample: $m = 8$, $a = 5$, $c = 1$, $X_0 = 0$.\nThe states are $0, 1, 6, 7, 4, 5, 2, 3, 0, \\ldots$, for instance $5 \\cdot 7 + 1 = 36 \\equiv 4 \\pmod 8$; all $8$ residues appear, so the period is $m$.\nNon-example: $X_{n+1} = X_n^2 \\bmod m$ is not a linear congruential generator, because the map $x \\mapsto x^2$ is not of the form $x \\mapsto ax + c$.\nThe map $x \\mapsto ax + c$ is an affine map of the ring of [[integers modulo $m$|03VFZ2WETT]], so the period is at most $m$.\nThe Hull–Dobell theorem treats $c \\ne 0$.\nThe period equals $m$ for every seed if and only if three conditions hold.\nThey are: $c$ and $m$ are coprime, every prime factor of $m$ divides $a - 1$, and $4$ divides $a - 1$ if $4$ divides $m$.\nIn the example, $\\gcd(1, 8) = 1$ and $a - 1 = 4$, so the conditions hold.\n",
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