We describe a new software package, named SumCracker, for proving and finding identities involving symbolic sums and related objects. SumCracker is applicable to a wide range of expressions for many of which there has not been any software available up to now. The purpose of this paper is to illustrate how to solve problems using that package.

. Monier and Rabin proved that an odd composite can pass the Strong Probable Prime Test for at most 1 4 of the possible bases. In this paper, a probable prime test is developed using quadratic polynomials and the Frobenius automorphism. The test, along with a fixed number of trial divisions, ensures that a composite n will pass for less than 1 7710 of the polynomials x 2 \Gamma bx \Gamma c with i b 2 +4c n j = \Gamma1 and \Gamma \Gammac n \Delta = 1. The running time of the test is asymptotically 3 times that of the Strong Probable Prime Test. x1 Background Perhaps the most common method for determining whether or not a number is prime is the Strong Probable Prime Test. Given an odd integer n, let n = 2 r s + 1 with s odd. Choose a random integer a with 1 a n \Gamma 1. If a s j 1 mod n or a 2 j s j \Gamma1 mod n for some 0 j r \Gamma 1, then n passes the test. An odd prime will pass the test for all a. The test is very fast; it requires no more than (1 +...

Abstract. The proliferation of probable prime tests in recent years has produced a plethora of definitions with the word “pseudoprime ” in them. Examples include pseudoprimes, Euler pseudoprimes, strong pseudoprimes, Lucas pseudoprimes, strong Lucas pseudoprimes, extra strong Lucas pseudoprimes and Perrin pseudoprimes. Though these tests represent a wealth of ideas, they exist as a hodge-podge of definitions rather than as examples of a more general theory. It is the goal of this paper to present a way of viewing many of these tests as special cases of a general principle, as well as to re-formulate them in the context of finite fields. One aim of the reformulation is to enable the creation of stronger tests; another is to aid in proving results about large classes of pseudoprimes. 1.

. We define a Carmichael number of order m to be a composite integer n such that nth-power raising defines an endomorphism of every Z=nZ-algebra that can be generated as a Z=nZ-module by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, and we give a heuristic argument (based on an argument of Erdos for the usual Carmichael numbers) that indicates that for every m there should be infinitely many Carmichael numbers of order m. The argument suggests a method for finding examples of higher-order Carmichael numbers; we use the method to provide examples of Carmichael numbers of order 2. 1. Introduction A Carmichael number is defined to be a positive composite integer n that is a Fermat pseudoprime to every base; that is, a composite n is a Carmichael number if a n j a mod n for every integer a. Clearly one can generalize the idea of a Carmichael number by allowing the pseudoprimality test in the definition to vary over some...

Abstract. We define a Carmichael number of order m to be a composite integer n such that nth-power raising defines an endomorphism of every Z/nZalgebra that can be generated as a Z/nZ-module by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, and we give a heuristic argument (based on an argument of Erdős for the usual Carmichael numbers) that indicates that for every m there should be infinitely many Carmichael numbers of order m. The argument suggests a method for finding examples of higher-order Carmichael numbers; we use the method to provide examples of Carmichael numbers of order 2. 1.

1.4 Simple XOR

this paper, we aim at presenting the most recent results achieved in the theory of pseudoprime numbers. First of all, we make a list of all pseudoprime varieties existing so far. This includes Lucas-pseudoprimes and the generalization to sequences generated by integer polynomials modulo N , elliptic pseudoprimes. We discuss the making of tables and the consequences on the design of very fast primality algorithms for small numbers. Then, we describe the recent work of Alford, Granville and Pomerance, in which they prove that there

Abstract. We prove the existence of infinitely many Perrin pseudoprimes, as conjectured by Adams and Shanks in 1982. The theorem proven covers a general class of pseudoprimes based on recurrence sequences. We use ingredients of the proof of the infinitude many Carmichael numbers, along with zero-density estimates for Hecke L-functions. 1. Background In a 1982 paper [1], Adams and Shanks introduced a probable primality test based on third order recurrence sequences. The following is a version of that test. Consider sequences An = An(r, s) defined by the following relations: A−1 = s, A0 = 3, A1 = r, and An = rAn−1 − sAn−2 + An−3. Let f(x) = x 3 − rx 2 + sx − 1 be the associated polynomial and ∆ its discriminant. (Perrin’s sequence is An(0, −1).) Definition. The signature mod m of an integer n with respect to the sequence Ak(r, s) is the 6-tuple (A−n−1, A−n, A−n+1, An−1, An, An+1) mod m. Definitions. An integer n is said to have an S-signature if its signature mod n is congruent to (A−2, A−1, A0, A0, A1, A2). An integer n is said to have a Q-signature if its signature mod n is congruent to (A, s, B, B, r, C), where for some integer a with f(a) ≡ 0 mod n, A ≡ a −2 + 2a, B ≡ −ra 2 + (r 2 − s)a, and C ≡ a 2 + 2a −1. An integer n is said to have an I-signature if its signature mod n is congruent to (r, s, D ′ , D, r, s), where D ′ + D ≡ rs − 3 mod n and (D ′ − D) 2 ≡ ∆. Definition. A Perrin pseudoprime with parameters (r, s) is an odd composite n such that either