The Euclidean Algorithm and the Linear Diophantine Equation ax plus by = gcd(a, b)

被引：2
|
作者：
Rankin, S. A. [1 ]
机构：
[1] Univ Western Ontario, Dept Math, London, ON N6A 5B7, Canada
来源：
AMERICAN MATHEMATICAL MONTHLY | 2013年 / 120卷 / 06期
关键词：
D O I：
10.4169/amer.math.monthly.120.06.562
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
In this note, we prove that for any positive integers a and b, with d = gcd(a, b), among all integral solutions to the equation ax + by = d, the solution (x(0), y(0)) that is provided by the Euclidean algorithm lies nearest to the origin. In fact, we prove that (x(0), y(0)) lies in the interior of the circle centered at the origin with radius 1/2d root a(2) + b(2).
引用
收藏
页码：562 / 564
页数：3
相关论文
共 50 条
  • [1] The Polynomial Euclidean Algorithm and the Linear Equation AX plus BY = gcd(A, B)
    Effinger, Gove
    Mullen, Gary L.
    MATHEMATICAL INTELLIGENCER, 2017, 39 (01): : 22 - 25
  • [2] The Polynomial Euclidean Algorithm and the Linear Equation AX + BY = gcd(A, B)
    Gove Effinger
    Gary L. Mullen
    The Mathematical Intelligencer, 2017, 39 : 22 - 25
  • [3] A study on the exponential Diophantine equation ax + (a plus b)y = bz
    Miyazaki, Takafumi
    Terai, Nobuhiro
    PUBLICATIONES MATHEMATICAE-DEBRECEN, 2019, 95 (1-2): : 19 - 37
  • [4] The restricted solutions of ax plus by = gcd(a, b)
    Lee, Ju-Si
    TAIWANESE JOURNAL OF MATHEMATICS, 2008, 12 (05): : 1191 - 1199
  • [5] On the Diophantine Equation ax
    Pakapongpun, Apisit
    Chattae, Bunthita
    INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 2022, 17 (02): : 711 - 716
  • [6] On the Diophantine Equation ax
    Dokchan, Rakporn
    Panngam, Nopparat
    INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 2025, 20 (01): : 63 - 66
  • [7] On the Diophantine equation ax
    Viriyapong, Chokchai
    Viriyapong, Nongluk
    INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 2023, 18 (03): : 525 - 527
  • [8] On the Diophantine equation ax
    Viriyapong, Chokchai
    Viriyapong, Nongluk
    INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 2024, 19 (02): : 449 - 451
  • [9] Diophantine equation Ax2+B = yn
    Cao, ZF
    Dong, XL
    CHINESE SCIENCE BULLETIN, 1998, 43 (13): : 1141 - 1142
  • [10] ON THE DIOPHANTINE EQUATION ax(3)
    Subburam, Sivanarayanapandian
    Thangadurai, Ravindrananathan
    FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI, 2015, 53 (01) : 167 - 175
12345