[1]薛晓果,孟倩倩,张 彬,等.丢番图方程ax+by=n的一个注记(英文)[J].南京师范大学学报(自然科学版),2015,38(04):32.
 Xue Xiaoguo,Meng Qianqian,Zhang Bin,et al.A Remark of Diophantine Equation ax+by=n[J].Journal of Nanjing Normal University(Natural Science Edition),2015,38(04):32.
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丢番图方程ax+by=n的一个注记(英文)()
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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38卷
期数:
2015年04期
页码:
32
栏目:
数学
出版日期:
2015-12-30

文章信息/Info

Title:
A Remark of Diophantine Equation ax+by=n
作者:
薛晓果1孟倩倩2张 彬3任福梅1
(1.南京师范大学数学科学学院,江苏 南京 210023)(2.兖州区东御桥小学,山东 兖州 272100)(3.曲阜师范大学数学科学学院,山东 曲阜 273165)
Author(s):
Xue Xiaoguo1Meng Qianqian2Zhang Bin3Ren Fumei1
(1.School of Mathematical Sciences,Nanjing Normal University,Nanjing 210023,China)(2.Dongyuqiao Primary School of Yanzhou District,Yanzhou 272100,China)(3.School of Mathematical Sciences,Qufu Normal University,Qufu 273165,China)
关键词:
丢番图方程生成函数留数定理
Keywords:
Diophantine equationgenerating functionResidue theorem
分类号:
11D45;11D04;O5A15
文献标志码:
A
摘要:
令[a,b]为互素的正整数,[n]为非负整数. [D(a,b;n)]表示不定方程[ax+by=n]的非负整数解[(x,y)]的个数. Tripathi证明了[D(a,b;n)=nab+121a+1b+1aj=1a-1ζ-jna1-ζbja+1bk=1b-1ζ-knb1-ζakb],其中[ζm=e2πi/m]. 在本文中,我们建立了[D(a,b;n)]的递推关系,从而给出了上述结论的新证明.
Abstract:
Let a,b be positive integers such that(a,b)=1 and let n be a non-negative integer. Define [D(a,b;n)] to be the number of non-negative integer solutions(x,y)of the Diophantine equation ax+by=n. Tripathi proved that[D(a,b;n)=nab+121a+1b+1aj=1a-1ζ-jna1-ζbja+1bk=1b-1ζ-knb1-ζakb],where [ζm=e2πi/m]. In this note,we put forward a recurrence relation of [D(a,b;n)],thus giving a new proof of above formula.

参考文献/References:

[1]DICKSON L E. History of the theory of numbers:diophantine analysis[M]. New York:Chelsea Publishing Co.,1966.
[2]HUA L K. Introduction to number theory[M]. Berlin:Springer-Verlag,1982.
[3]NIVEN I,ZUCKERMAN H S,MONTGOMERY H L. An introduction to the theory of numbers[M]. 5th ed. New York:John Wiley & Sons,Inc,1991.
[4]PONNUSAMY S,SILVERMAN H. Complex variables with applications[M]. Berlin:Birkhauser Boston,2006.
[5]TRIPATHI A. The number of solutions to [ax+by=n][J]. Fibonacci quart,2000(38):290-293.
[6]WILF H S. Generating function ology[M]. 3rd ed. Wellesley:A K Peters,Ltd,2006.

相似文献/References:

[1]马米米,吴建东.关于丢番图方程(65n)x+(72n)y=(97n)z[J].南京师范大学学报(自然科学版),2014,37(04):28.
 Ma Mimi,Wu Jiandong.On the Diophantine Equation(65n)x+(72n)y=(97n)z[J].Journal of Nanjing Normal University(Natural Science Edition),2014,37(04):28.

备注/Memo

备注/Memo:
Received data:2015-09-17. 
Foundation item:Supported by Project of Graduate Education Innovation of Jiangsu Province(KYLX_0690),Research Fund for the
Doctoral Program of Higher Education of China(20133207110012)and the Doctoral Starting up Foundation of Qufu Normal University. Corresponding author:Zhang Bin,lecturer,majored in algebraic number theory. E-mail:zhangbin100902025@163.com
更新日期/Last Update: 2015-12-30