but rather select them in a certain way depending on the input number n in order to decrease the proportion of false positives, i.e., composite numbers that pass the test. Sometimes such composite numbers are commonly called Frobenius pseudoprimes, although they may correspond to different parameters. Using parameter selection ideas first laid out in Baillie and Wagstaff (1980) as part of the Baillie–PSW primality test and used by Grantham in his quadratic Frobenius test, one can create even better quadratic tests. In particular, it was shown that choosing parameters from quadratic non-residues modulo n (based on the Jacobi symbol) makes far stronger tests, and is one reason for the success of the Baillie–PSW primality test. For instance, for the parameters (P,2), where P is the first odd integer that satisfies , there are no pseudoprimes below . Yet another test is proposed by Khashin. For a given non-square number n, it first computes a parameter c as the smallest odd prime having Jacobi symbol , and then verifies the congruence: . While all prime n pass this test, a composite n passes it if and only if n is a Frobenius pseudoprime for . Similar to the above example, Khashin notes that no pseudoprime has been found for his test. He further shows that any that exist under 260 must have a factor less than 19 or have c > 128. Properties The computational cost of the Frobenius pseudoprimality test with respect to quadratic polynomials is roughly three times the cost of a strong pseudoprimality test (e.g. a single round of the Miller–Rabin primality test), 1.5 times that of a Lucas pseudoprimality test, and slightly more than a Baillie–PSW primality test. Note that the quadratic Frobenius test is stronger than the Lucas test. For example, 1763 is a Lucas pseudoprime to (P, Q) = (3, -1) since U1764(3,-1) ≡ 0 (mod 1763) (U(3,-1) is given in ), and it also passes the Jacobi step since , but it fails the Frobenius test to x2 - 3x - 1. This property can be clearly seen when the algorithm is formulated as shown in Crandall and Pomerance Algorithm 3.6.9 or as shown by Loebenberger, as the algorithm does a Lucas test followed by an additional check for the Frobenius condition. While the quadratic Frobenius test does not have formal error bounds beyond that of the Lucas test, it can be used as