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  "body": "\n## Example\n\nFix integers $P$ and $Q$.\nThe *Lucas sequences* $U_n(P, Q)$ and $V_n(P, Q)$ are the [[solutions|06MJC8YDYR]] of the recurrence\n$$x_{n+2} = P\\, x_{n+1} - Q\\, x_n \\qquad (n \\ge 0)$$\nwith initial values $U_0 = 0$, $U_1 = 1$ and $V_0 = 2$, $V_1 = P$.\nThe [[characteristic polynomial|0YA7X86K05]] is $X^2 - P X + Q$.\nIf its discriminant $D = P^2 - 4Q$ is not $0$, let $a, b$ be its two distinct complex roots; then\n$$U_n = \\frac{a^n - b^n}{a - b}, \\qquad V_n = a^n + b^n.$$\nCheck: both formulas solve the recurrence, since $a^{n+2} = P a^{n+1} - Q a^n$ and likewise for $b$, and they give $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = a + b = P$.\n\nSpecial cases:\n\n1. $P = 1$, $Q = -1$: $U_n$ are the [[Fibonacci numbers|0GA9K57A39]] $0, 1, 1, 2, 3, 5, \\dots$ and $V_n$ are the Lucas numbers $2, 1, 3, 4, 7, 11, \\dots$.\n2. $P = 2$, $Q = -1$: $U_n$ are the [[Pell numbers|JY4ZN779YG]] $0, 1, 2, 5, 12, 29, \\dots$.\n3. $P = 3$, $Q = 2$: $U_n = 2^n - 1$, the Mersenne numbers $0, 1, 3, 7, 15, \\dots$, since the roots are $2$ and $1$.\n\n## Explanations\n\n### L1 human\n\nA Lucas sequence is a two-parameter family that contains the Fibonacci numbers and many of their relatives.\nEach term is $P$ times the previous term minus $Q$ times the one before.\nChanging $P$ and $Q$ gives the Fibonacci numbers, the Pell numbers, or the numbers $2^n - 1$.\nBecause they share one formula, facts proved once for Lucas sequences hold for all of these at the same time.\nThey are used in primality tests, for example the [[Lucas–Lehmer test|061BTYA0RN]] for [[Mersenne primes|0F3Z22M2GT]].\n",
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