Fibonacci polynomial are determined in terms of the Fibonacci numbers and Lucas numbers . Such Frobenius pseudoprimes form the sequence: 4181, 5777, 6721, 10877, 13201, 15251, 34561, 51841, 64079, 64681, 67861, 68251, 75077, 90061, 96049, 97921, 100127, 113573, 118441, 146611, 161027, 162133, 163081, 186961, 197209, 219781, 231703, 252601, 254321, 257761, 268801, 272611, 283361, 302101, 303101, 330929, 399001, 430127, 433621, 438751, 489601, ... . While 323 is the first Lucas pseudoprime with respect to the Fibonacci polynomial , the first Frobenius pseudoprime with respect to the same polynomial is 4181 (Grantham stated it as 5777 but multiple authors have noted this is incorrect and is instead the first pseudoprime with for this polynomial). Another case, Frobenius pseudoprimes with respect to the quadratic polynomial can be determined using the Lucas sequence and are: 119, 649, 1189, 4187, 12871, 14041, 16109, 23479, 24769, 28421, 31631, 34997, 38503, 41441, 48577, 50545, 56279, 58081, 59081, 61447, 75077, 91187, 95761, 96139, 116821, 127937, 146329, 148943, 150281, 157693, 170039, 180517, 188501, 207761, 208349, 244649, 281017, 311579, 316409, 349441, 350173, 363091, 371399, 397927, 423721, 440833, 459191, 473801, 479119, 493697, ... In this case, the first Frobenius pseudoprime with respect to the quadratic polynomial is 119, which is also the first Lucas pseudoprime with respect to the same polynomial. Besides, . The quadratic polynomial , i.e. , has sparser pseudoprimes as compared to many other simple quadratics. Using the same process as above, we get the sequence: 13333, 44801, 486157, 1615681, 3125281, 4219129, 9006401, 12589081, 13404751, 15576571, 16719781, …. Notice there are only 3 such pseudoprimes below 500000, while there are many Frobenius (1, −1) and (3, −1) pseudoprimes below 500000. Every entry in this sequence is a Fermat pseudoprime to base 5 as well as a Lucas (3, −5) pseudoprime, but the converse is not true: 642001 is both a psp-5 and a Lucas (3,-5) pseudoprime, but is not a Frobenius (3, −5) pseudoprime. (note that Lucas pseudoprime for a pair need not to be a Fermat pseudoprime for base ||, e.g. 14209 is a Lucas (1, −3) pseudoprime, but not a Fermat pseudoprime for base 3. Strong Frobenius pseudoprimes Strong Frobenius pseudoprimes are also defined. Details on implementation for quadratic polynomials can be found in Crandall and Pomerance. Pseudoprimality tests The conditions defining Frobenius pseudoprime can be used for testing a given number n for probable primality. Often such tests do not rely on fixed parameters ,