Lucas pseudoprime

Lucas pseudoprimes and Fibonacci pseudoprimes are composite integers that pass certain tests which all primes and very few composite numbers pass: in this case, criteria relative to some Lucas sequence.

Given integers P and Q, where P > 0 and ${\displaystyle D=P^{2}-4Q}$, let U_k(P, Q) and V_k(P, Q) be the corresponding Lucas sequences.

Let n be a positive integer and let ${\displaystyle \left({\tfrac {D}{n}}\right)}$ be the Jacobi symbol. We define

If n is a prime such that the greatest common divisor of n and Q (that is, GCD(n, Q)) is 1, then the following congruence condition holds:

If this equation does not hold, then n is not prime. If n is composite, then this equation usually does not hold. These are the key facts that make Lucas sequences useful in primality testing.

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