Localized Factorizations of Integers
نویسنده
چکیده
We determine the order of magnitude of H(k+1)(x,y, 2y), the number of integers n ≤ x that are divisible by a product d1 · · · dk with yi < di ≤ 2yi, when the numbers log y1, . . . , log yk have the same order of magnitude and k ≥ 2. This generalizes a result by Kevin Ford when k = 1. As a corollary of these bounds, we determine the number of elements up to multiplicative constants that appear in a (k + 1)-dimensional multiplication table as well as how many distinct sums of k + 1 Farey fractions there are modulo 1.
منابع مشابه
Delange’s Tauberian theorem and asymptotic normality of random ordered factorizations of integers
By a suitable shifting-the-mean parametrization at the Dirichlet series level and Delange’s Tauberian theorems, we show that the number of factors in random ordered factorizations of integers is asymptotically normally distributed.
متن کامل#A16 INTEGERS 12A (2012): John Selfridge Memorial Issue THE SEARCH FOR AURIFEUILLIAN-LIKE FACTORIZATIONS
We searched the Cunningham tables for new algebraic factorizations similar to those discovered by Aurifeuille. A naive search would have been too slow. We accelerated it enough to make it feasible. Many interesting results were found. –Dedicated to the memory of John Selfridge, who loved the integers.
متن کاملA central limit theorem for random ordered factorizations of integers
Write an integer as finite products of ordered factors belonging to a given subset P of integers larger than one. A very general central limit theorem is derived for the number of ordered factors in random factorizations for any subset P containing at least two elements. The method of proof is very simple and relies in part on Delange’s Tauberian theorems and an interesting Tauberian technique ...
متن کاملOrdered and Unordered Factorizations of Integers
We study the number of ways of writing a positive integer n as a product of integer factors greater than one. We survey methods from the literature for enumerating and also generating lists of such factorizations for a given number n. In addition, we consider the same questions with respect to factorizations that satisfy constraints, such as having all factors distinct. We implement all these m...
متن کاملNon-unique Factorization and Principalization in Number Fields
Following what is basically Kummer’s relatively neglected approach to non-unique factorization, we determine the structure of the irreducible factorizations of an element n in the ring of integers of a number field K. Consequently, we give a combinatorial expression for the number of irreducible factorizations of n in the ring. When K is quadratic, we show in certain cases how quadratic forms c...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009