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A Remark of Diophantine Equation ax+by=n(PDF)

《南京师大学报（自然科学版）》[ISSN:1001-4616/CN:32-1239/N]

Issue:
2015年04期
Page:
32-
Research Field:
数学
Publishing date:

Info

Title:
A Remark of Diophantine Equation ax+by=n
Author(s):
Xue Xiaoguo1; Meng Qianqian2; Zhang Bin3; Ren 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）
Keywords:
Diophantine equation; generating function; Residue theorem
PACS:
11D45；11D04；O5A15
DOI:
-
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.

References:

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［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.

Memo

Memo:
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Last Update: 2015-12-30