[1]方金辉.弱素性可加数性质的研究[J].南京师范大学学报(自然科学版),2018,41(04):26.[doi:10.3969/j.issn.1001-4616.2018.04.005]
　Fang Jinhui.Note on the Weakly Prime-Additive Numbers[J].Journal of Nanjing Normal University(Natural Science Edition),2018,41(04):26.[doi:10.3969/j.issn.1001-4616.2018.04.005]
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弱素性可加数性质的研究()
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《南京师范大学学报》（自然科学版）[ISSN:1001-4616/CN:32-1239/N]

卷:
第41卷
期数:
2018年04期
页码:
26
栏目:
·数学与计算机科学·
出版日期:
2018-12-31

文章信息/Info

Title:
Note on the Weakly Prime-Additive Numbers
文章编号:
1001-4616(2018)04-0026-03
作者:
方金辉
南京信息工程大学数学与统计学院,南京 江苏 210044
Author(s):
Fang Jinhui
School of Mathematics and Statistics,Nanjing University of Information Science and Technology,Nanjing 210044,China
关键词:
弱素性可加数; Dirichlet定理; 中国剩余定理
Keywords:
weakly prime-additive numbers; Dirichlet’s theorem; Chinese remainder theorem
分类号:
O156.1
DOI:
10.3969/j.issn.1001-4616.2018.04.005
文献标志码:
A
摘要:
设n是正整数,若n有至少两个互异素因子,而且存在n的互异素因子p1,p2,…,pt和正整数α1,α2,…,αt使得n=pα11+pα22+…+pαtt,那么我们称n为弱素性可加数.本文中,我们通过多次巧妙应用中国剩余定理、Dirichlet定理和二次互反律证明:对任意正整数m和t,存在无穷多个弱素性可加数n使得m|n且n=pα11+pα22+…+pα4t4t+pα4t+14t+1,其中p1,p2,…,p4t+1是n的互异素因子,α1,α2,…,α4t+1是正整数.
Abstract:
A positive integer n is called weakly prime-additive if n has at least two distinct prime divisors and there exist distinct prime divisors p1,p2,…,pt of n and positive integers α1,α2,…,αt such that n=pα11+pα22+…+pαtt. In this paper,by employing Chinese remainder theorem,Dirichlet’s theorem and the quadratic reciprocity law,we prove that,for any positive integers m and t,there exist infinitely many weakly prime-additive numbers n with m|n and n=pα11+pα22+…+pα4t4t+pα4t+14t+1,where p1,p2,…,p4t+1 are distinct prime divisors of n and α1,α2,…,α4t+1 are positive integers.

参考文献/References:

[1] ERDOS P,HEGYVARI N. On prime-additive numbers[J]. Studia Sci Math Hungar,1992,27:207-212.
[2]FANG J H,CHEN Y G. On the shortest weakly prime-additive numbers[J]. J number theory,2018,182:258-270.
[3]潘承洞,潘成彪. 初等数论[M]. 3版. 北京大学出版社,2013.
[4]NATHANSON M B. Elementary Methods in Number Theory[M]. New York:Springer,2000.

备注/Memo

备注/Memo:
收稿日期:2018-03-15.
基金项目:国家自然科学基金(11671211).
通讯联系人:方金辉,博士,副教授,研究方向:数论研究. E-mail:fangjinhui1114@163.com
更新日期/Last Update: 2018-12-30