Please wait a minute...
浙江大学学报（理学版）
浙江大学学报（理学版）  2016, Vol. 43 Issue (2): 138-143    DOI: 10.3785/j.issn.1008-9497.2016.02.003
数学与计算机科学     
关于丢番图方程X2-(a2+1)Y4=35-12a的讨论
管训贵
泰州学院数学系, 江苏泰州 225300
Discussion on the Diophantine equation X2-(a2+1)Y4=35-12a
GUAN Xungui
Department of Mathematics, Taizhou University, Taizhou 225300, Jiangsu Province, China
 全文: PDF(1318 KB)  
摘要： 设a是正整数,证明了当a=1时,方程X2-(a2+1)Y4=35-12a仅有正整数解(X,Y)=(5,1);当a=2时,该方程仅有正整数解(X,Y)=(4,1)和(56,5);当a=3时,该方程仅有正整数解(X,Y)=(3,1);当a=4时,该方程仅有正整数解(X,Y)=(2,1)和(202,7);当a=5时,该方程仅有1组互素的正整数解(X,Y)=(1,1);当a=6时,该方程无正整数解(X,Y);当a≥7且12a+1为非平方数时,该方程最多有3组互素的正整数解(X,Y);当a≥7且12a+1为平方数时,该方程最多有4组互素的正整数解(X,Y).
关键词： 四次方程;  虚二次域;  丢番图逼近;  解数;  上界    
Abstract: Let a be an positive integer. We prove that if a=1, then the equation X2-(a2+1)Y4=35-12a has only one positive integer solution (X, Y)=(5, 1); If a=2, then the equation has only two positive integer solutions, (X, Y)=(4, 1) and (56, 5); If a=3, then the equation has only one positive integer solution (X, Y)=(3, 1); If a=4, then the equation has two positive integer solutions (X, Y)=(2, 1) and (202, 7); If a=5, then the equation has one coprime positive integer solution (X, Y)=(1, 1); If a=6, then the equation has no positive integer solution (X, Y); If a≥7 and 12a+1 is a nonsquare positive integer, the equation has at most three coprime positive integer solutions; While if a≥7 and 12a+1 is a square, the equation has at most four coprime positive integer solutions.
Key words: quartic equations    imaginary quadratic fields    Diophantine approximations    number of positive integer solutions    upper bound
收稿日期: 2015-06-01 出版日期: 2016-03-12
CLC:  O156.7  
基金资助: 江苏省教育科学"十二五"规划项目(D201301083);云南省教育厅科研项目(2014Y462);泰州学院教授基金项目(TZXY2015JBJJ002).
作者简介: 管训贵(1963-),ORCID:http://orcid.org/0000-0001-7612-2635,男,本科,教授,主要从事数论研究,E-mail:tzszgxg@126.com.
服务　　
把本文推荐给朋友
加入引用管理器
E-mail Alert
RSS
作者相关文章　　
管训贵

引用本文:

管训贵. 关于丢番图方程X2-(a2+1)Y4=35-12a的讨论[J]. 浙江大学学报（理学版）, 2016, 43(2): 138-143.

GUAN Xungui. Discussion on the Diophantine equation X2-(a2+1)Y4=35-12a. Journal of Zhejiang University (Science Edition), 2016, 43(2): 138-143.

链接本文:

https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2016.02.003    或    https://www.zjujournals.com/sci/CN/Y2016/V43/I2/138

[1] LJUNGGREN W. Zur theorie der Gleichung x2+1=Dy4[J].Avh Norsk Vid Akad Oslo,1942(5):1-27.
[2] LJUNGGREN W. On the Diophantine equation Ax4-By2=C(C=1,4)[J].Math Scand,1967,21(2):149-158.
[3] 袁平之,张中锋.丢番图方程X2-(a2+4p2n)Y4=-4p2n[J].数学学报:中文版,2014,57(2):209-222. YUAN Pingzi, ZHANG Zhongfeng. On the Diophantine equation X2-(a2+4p2n)Y4=-p2n[J]. Acta Mathematica Sinica:Chinese Series,2014,57(2):209-222.
[4] STOLL M, WALSH P G, YUAN P Z. On the Diophantinc equation X2-(a2m+1)Y4=22m[J]. Acta Arith,2009,139(1):57-63.
[5] YUAN P Z. Squares in Lehmer sequences and the Diophantine equation Ax4-By2=2[J]. Acta Arith,2009,139(3):275-302.
[6] YUAN P Z, ZHANG Z F. On the Diophantine equation X2-(1+a2)Y4=-2a[J]. Sci China:Ser A,2010,53(8):2143-2158.
[7] 柯召,孙琦.谈谈不定方程[M].上海:上海教育出版社,1980:30-31. KE Zhao, SUN Qi. On the Diophantine Equations[M]. Shanghai:Shanghai Education Press, 1980:30-31.
[8] CHEN J H, VOUTIER P M. A complete solution of the Diophantine equation x2+1=dy4 and a related family of quartic Thue equations[J]. J Number Theory,1997,62(1):71-99.
[9] COHN J H E. Some quartic Diophantine equations[J]. Pacific J Math, 1968,26:233-243.
[10] HUA L K, Introduction to Number Theory[M]. Translated by PETER S. Berlin, New York:Springer-Verlag,1982.
[11] DUJELLA A, Continued fractions and RSA with small secret exponent[J]. Tatra Mt Math Publ,2004,29:101-112.
[12] 孙琦,袁平之.关于不定方程x4-Dy2=1的一个注记[J].四川大学学报:自然科学版,1997,34(3):265-268. SUN Qi, YUAN Pingzhi. On the Diophantine equation x4-Dy2=1[J]. Journal of Sichuan University:Natural Science Edition,1997,34(3):265-268.
[1] 周后卿. 卡氏积图的Laplacian谱半径的上界[J]. 浙江大学学报（理学版）, 2018, 45(1): 10-13,17.