无穷递降法在不定方程中的应用  

Application of the Method of Infinite Descent to Indefinite Equations

作　　者：管训贵 GUAN Xun-gui(School of Mathematics and Physics,Taizhou University,Taizhou 225300,China)

机构地区：[1]泰州学院数理学院,江苏泰州225300

出　　处：《广西师范学院学报（自然科学版）》2018年第1期24-27,共4页Journal of Guangxi Teachers Education University(Natural Science Edition)

基　　金：国家自然科学基金(11471144);江苏省自然科学基金(BK20171318);泰州学院教博基金(TZXY2016JBJJ001)

摘　　要：利用无穷递降法证明了:(1)若素数p=48 m+41(m≥0),则不定方程x^4+3py^4=z^2(y≠0)无整数解;(2)不定方程x^4+4x^3y-6x^2y^2-4xy^3+y^4=z^2的全部正整数解可表为(x_n,y_n,z_n)=(K_nd_n,L_nc_n,K_n^2c_n^2-2L_n^2d_n^2),这里Ln/Kn=cndn±en/c2n+2d2n(cndn>en),dn,cn,en满足2d_n^4-c_n^4=e_n^2.It has been proven by the method of infinite descent that:(1)if p be a prime with p=48m+41(m≥0),then the Diophantine equation x 4+3py 4=z 2(y≠0)has no integer solution;(2)all positive integral solutions of the Diophantine equation x 4+4x 3y-6x 2y 2-4xy 3+y 4=z 2 are(x n,y n,z n)=(K nd n,L nc n,K 2 nc 2 n-2L 2 nd 2 n),where L n K n=c nd n±e n c 2 n+2d 2 n(c nd n>e n),and d n,c n,e n satisfied 2d 4 n-c 4 n=e 2 n.

关 键 词：无穷递降法 不定方程 整数解 模 同余 

分 类 号：O156.7[理学—数学]