[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）(
)
《南京师范大学学报》（自然科学版）[ISSN:1001-4616/CN:32-1239/N]
- 卷:
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第38卷
- 期数:
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2015年04期
- 页码:
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32
- 栏目:
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数学
- 出版日期:
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2015-12-30
文章信息/Info
- Title:
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A Remark of Diophantine Equation ax+by=n
- 作者:
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薛晓果1; 孟倩倩2; 张 彬3; 任福梅1
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（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):
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Xue Xiaoguo1; Meng Qianqian2; Zhang Bin3; Ren Fumei1
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（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）
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- 关键词:
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丢番图方程; 生成函数; 留数定理
- Keywords:
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Diophantine equation; generating function; Residue theorem
- 分类号:
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11D45；11D04；O5A15
- 文献标志码:
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A
- 摘要:
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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:
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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:
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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