TAILIEUCHUNG - Báo cáo toán học: "ON PRIMITIVE 3-SMOOTH PARTITIONS O"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: ON PRIMITIVE 3-SMOOTH PARTITIONS O. | ON PRIMITIVE 3-SMOOTH PARTITIONS OF n Michael R. Avidgn Submitted September 30 1996 Accepted December 2 1996 This paper is dedicated to Paul Erdos on the sad occasion of his recent death. Abstract. a primitive 3-smooth partition of n is a representation of n as the sum of numbers of the form 2a3b where no summand divides another. Partial results are obtained in the problem of determining the maximal and average order of the number of such representations. Results are also obtained regarding the size of the terms in such a representation resolving questions of Erdos and Selfridge. 0. Introduction Recently Erdos proposed the following problem let r n be the number of representations of n as the sum of 3-smooth numbers integers of the form 2a3b with a b 0 which are primitive no summand divides another . Determine i. the maximal order of r n ii. the average order of r n . It is easy to show that r n 1 for all n see 1 2 . In this paper partial results are obtained for both of these problems. Specihcally we prove Theorem 1. For n 5 r n n where a log2 log3 . Theorem 2. Let R x Vn x r n . Then x _. . x x logx 3 2 where 0 i ũk log . log 3 log 2 log 2 log 3 Dehne g n as the maximum over all representations of the minimum term . 11 8 3 9 2 so g 11 3 . Erdos has asked if limn 1 g n 1. We answer this in the affirmative by proving 1991 Mathematics Subject Classification. Primary 11P85 Secondary 05A17. Typeset by AmS-TEX 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 4 1997 R2 2 Theorem 3. g n 3 n where logl . Concerning g n Selfridge has asked if for each n g n appears as the least term in exactly one representation of n. We answer this in the negative by providing an inhnite number of counter-examples where it appears in two representations. The author would like to thank Professor Carl Pomerance for suggesting these problems and also for his counsel as the work progressed. 1. Proof of THEOREm 1 Lemma 1. r 2a3bm r m Proof. It is enough to show that r 2m r

TÀI LIỆU LIÊN QUAN
Đã phát hiện trình chặn quảng cáo AdBlock
Trang web này phụ thuộc vào doanh thu từ số lần hiển thị quảng cáo để tồn tại. Vui lòng tắt trình chặn quảng cáo của bạn hoặc tạm dừng tính năng chặn quảng cáo cho trang web này.