In combinatorial number theory, zero-sum problems, subset sums and covers of the integers are three different topics initiated by P. Erdös and investigated by many researchers; they play important roles in both number theory and combinatorics. In this paper we announce some deep connections among these seemingly unrelated fascinating areas, and aim at establishing a unified theory! Our main theorem unifies many results in these three realms and also has applications in many aspects such as finite fields and graph theory. To illustrate this, here we state our extension of the Erdös-Ginzburg-Ziv theorem: If A = {as(mod ns)} k s=1 covers some integers exactly 2p − 1times and others exactly 2p times, where p is a prime, then for any c1, ·· ·,ck ∈ Z/pZ there exists an I ⊆{1, ·· ·,k} such that � s∈I 1/ns = p and s∈I cs =0.

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].

Zero-sum problems on abelian groups, subset sums in a field and covers of the integers by residue classes, are three different active topics initiated by P. Erdős and investigated by many researchers. In an earlier

Numbers of the form (6m + 1)(12m + 1)(18m + 1) where all three factors are simultaneously prime are the best known examples of Carmichael numbers. In this paper we tabulate the counts of such numbers up to 10 for each n 42. We also derive a function for estimating these counts that is remarkably accurate.

: Goldach's conjecture states that every even integer greater or equal to 6 is the sum of two prime numbers. This result is still unproved. This conjecture has been numerically checked up to 4:10 11 on an IBM 3083 mainframe. We describe here an implementation on a less powerful machine which raises the bound to 10 12 . Key-words: Prime numbers; Goldbach's problem (R'esum'e : tsvp) CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE Centre National de la Recherche Scientifique Institut National de Recherche en Informatique (URA 227) Universit e de Rennes 1 -- Insa de Rennes et en Automatique -- unit e de recherche de Rennes Am'eliorations de bornes au sujet de la conjecture de Goldbach (premi`ere version) R'esum'e : La conjecture de Goldbach stipule que tout nombre pair sup'erieur ou 'egal `a 6 est somme de deux nombres premiers. Ce r'esultat est `a ce jour non d'emontr'e. Il a 'et'e v'erifi'e num'eriquement jusqu'`a 4:10 11 sur un IBM 3083. Nous d'ecrivons ici une impl'ementation...

in celebration of his Sixtieth Birthday Let d be a square-free integer, which may be positive or negative, and let h(−d) be the class number of Q ( √ −d). In this paper we investigate the frequency of values of d for which 3|h(−d). It follows from conjectures of Cohen and Lenstra [3], that asymptotically a constant proportion of values of d have this property. The conjectured proportion is different for positive and negative d, being 1 − (1 − 3 −j) j=1 in the case of imaginary quadratics, for example. It follows from the work of Davenport and Heilbronn [5] that a positive proportion of d have 3 ∤ h(−d), both in the case of d positive and d negative. However it remains an open problem whether or not the same is true for values with 3|h(−d). Write N−(X) for the number of positive square-free d ≤ X for which 3|h(−d), and similarly let N+(X) be the number of positive square-free d ≤ X for which 3|h(d). It was shown by Ankeny and Chowla [1] that N−(X) tends to infinity with X, and in fact their method yields N−(X) ≫ X 1/2. The best known result in this direction is that due to Soundararajan [7], who shows that N−(X) ≫ε X 7/8−ε, for any positive ε. In the case of real quadratic fields it was shown by Byeon and Koh [2] how Soundararajan’s analysis can be adapted to prove N+(X) ≫ε X 7/8−ε. The purpose of this note is to present a small improvement on these results, as follows.

Abstract not found

Abstract. In this paper, we introduce a certain method to construct polynomials producing many absolute pseudoprimes. By this method, we give new polynomials producing absolute pseudoprimes with any fixed number of prime factors which can be viewed as a generalization of Chernick’s result. By the similar method, we give another type of polynomials producing many absolute pseudoprimes. As concrete examples, we tabulate the counts of such numbers of our forms. 1.