A Note on Strong Dickson Pseudoprimes

Abstract

In 1994 the infinitude of Fermat pseudoprimes to any bases (i.e. Carmichael numbers) has been established (cf. [2, 42]), the smallest being 561 = 3 ċ 11 ċ 17. If, instead of the power function the Dickson polynomial, respectively Lucas sequence, V _n (P, Q) ≡P (mod n), is being used as the primality testing function for P, Q∈ℤ, then the smallest pseudoprime to this test with respect to all parameters P and Q is 443372888629441 = 17 ċ 31 ċ 41 ċ 43 ċ 89 ċ 97 ċ 167 ċ 331. Not more than about 50 of such strong Dickson pseudoprimes are known at present, each of them having a more complex structure than the (Fermat-) Carmichael numbers. Those particular properties will be the focus of our attention in this note. After summarizing some results on strong Dickson pseudoprimes their connections to other types of pseudoprimes are described and characterizations of superstrong Dickson pseudoprimes are presented. Furthermore, an algorithm for an effective generation of those kinds of numbers is introduced.