A prototype of zero-sum theorems, the well-known theorem of Erdős, Ginzburg and Ziv says that for any positive integer n, any sequence a1,a2,…,a2n-1 of 2n-1 integers has a subsequence of n elements whose sum is 0 modulo n. Appropriate generalizations of the question, especially that for (Z/pZ)d, generated a lot of research and still have challenging open questions. Here we propose a new generalization of the Erdős–Ginzburg–Ziv theorem and prove it in some basic cases
We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection which correspond to curves over prime fields or to curves with a prescribed torsion. Some of our results are rigorous and are based on recent advances in analytic number theory, some are conditional under certain widely believed conjectures, and others are purely heuristic in nature.
Let 1 be a finitely generated subgroup of Q* with rank r. We study the size of the order |1 p | of 1 mod p for density-one sets of primes. Using a result on the scarcity of primes p x for which p&1 has a divisor in an interval of the type [ y, y exp log { y] ({t0.15), we deduce that |1 p | p rÂ(r+1) exp log { p for almost all p and, assuming the Generalized Riemann Hypothesis, we show that |1 p | p ( p) ( Ä ) for almost all p. We also apply this to the Brown Zassenhaus Conjecture concerned with minimal sets of generators for primitive roots.
AcademicPress, Inc.
We obtain an asymptotic formula for the number of square-free values among p À 1; for primes ppx; and we apply it to derive the following asymptotic formula for LðxÞ; the number of square-free values of the Carmichael function lðnÞ for 1pnpx; LðxÞ ¼ ðk þ oð1ÞÞ x ln 1Àa x ; where a ¼ 0:37395y is the Artin constant, and k ¼ 0:80328y is another absolute constant.
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