Erdös [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance [2] showed that there are more than x 2=7 Carmichael numbers up to x, and gave arguments which even convinced Shanks (in person-to-person discussions) that Erdös must be correct. Nonetheless, Shanks's skepticism stemmed from an appropriate analysis of the data available to him (and his reasoning is still borne out by Pinch's extended new data [14,15]), and so we herein derive conjectures that are consistent with Shanks's observations, while tting in with the viewpoint of Erdös [8] and the results of [2,3].

Abstract For many years mathematicians have searched for a fast and reliable primality test. This is especially relevant nowadays, because the RSA public-key cryptosystem requires very large primes in order to generate secure keys. I will describe some efficient randomised algorithms that are useful in practice, but have the defect of occasionally giving the wrong answer, or taking a very long time to give an answer. Recently Agrawal, Kayal and Saxena found a deterministic polynomial-time primality test. I will describe their algorithm, mention some improvements by Bernstein and Lenstra, and explain why this is not the end of the story.

A Carmichael number N is a composite number N with the property that for every b prime to N we have b N−1 ≡ 1 mod N. For background on Carmichael numbers and details of previous computations see [1]: in that paper we described the computation of the Carmichael numbers up to 10 15 and presented some statistics. These computations have since been extended to 10 18 [2] and now to 10 19, using similar techniques, and we present further statistics. The principal search was a depth-first back-tracking search over possible sequences of primes factors p1,..., pd satisfying pi coprime to pj(pj − 1) for i � = j. We also employed the variant based on proposition 2 of [1] which determines the finitely many possible pairs (pd−1, pd) from Pd−2. In practice this was useful only when d = 3. Finally we employed a different search over large values of pd, in the range 2.10 6 < pd < 10 9.5, using the property that Pd−1 ≡ 1 mod (pd − 1). We have shown that there are 3381806 Carmichael numbers up to 10 19, all with at most 12 prime factors. We let C(X) denote the number of Carmichael numbers

We consider the classical problem of testing if a given (large) number n is prime or composite. First we outline some of the efficient randomised algorithms for solving this problem. For many years it has been an open question whether a deterministic polynomial time algorithm exists for primality testing, i.e. whether "PRIMES is in P". Recently Agrawal, Kayal and Saxena answered this question in the affirmative. They gave a surprisingly simple deterministic algorithm. We describe their algorithm, mention some improvements by Bernstein and Lenstra, and consider whether the algorithm is useful in practice. Finally, as a topic for future research, we mention a conjecture that, if proved, would give a fast and practical deterministic primality test.

We extend our previous computations to show that there are 8220777 Carmichael numbers up to 10 20. As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a “large prime variation”. We present further statistics on the distribution of Carmichael

Recall that Fermat’s “little theorem ” says that if p is prime and a is not a multiple of p, then ap−1 ≡ 1 mod p. This theorem gives a possible way to detect primes, or more exactly, non-primes: if for some positive a ≤ n − 1, an−1 is not congruent to 1 mod n, then, by the theorem, n is

Abstract. We extend our previous computations to show that there are 20138200 Carmichael numbers up to 1021. As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a “large prime variation”. We present further statistics on the distribution of Carmichael numbers. 1.

Abstract. We describe some primality tests based on quadratic rings and discuss the absolute pseudoprimes for these tests. 1.

We extend our previous computations to show that there are 20138200 Carmichael numbers up to 10 21. As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a “large prime variation”. We present further statistics on the distribution of Carmichael

Abstract. We extend our previous computations to show that there are 246683 Carmichael numbers up to 10 16. As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a “large prime variation”. We present further statistics on the distribution of Carmichael numbers. 1.