【作者】 李伟；

【导师】 刘旭；

【作者基本信息】 兰州交通大学 ， 运筹学与控制论， 2015， 硕士

【摘要】 丢番图方程是指未知数个数多于方程个数且取整数值的方程(或方程组),是数论中一个很重要的内容和研究课题,与代数数论、组合数学、代数几何等有密切联系。它的研究成果不仅对数学各个分支的发展起着重要作用,而且对其它学科如物理学、经济学、计算机科学等有很大的应用价值。因此,丢番图方程一直是众多数学工作者热衷研究的对象。本文的主要内容为:1.论述了丢番图方程的概况、丢番图方程的主要成就、解丢番图方程的困难性以及求解原则。2.给出了本文的预备知识,包括同余理论、二次剩余、Legendre符号、Pell方程的一些主要相关定义、性质、定理等。3.介绍了丢番图方程Ax2+B=yn的研究进展,并用代数数论的方法证明了丢番图方程Ax2+B=yn在(A,B,n)=(1,4,9)时无整数解。4.介绍了丢番图方程x2-Dy4=N的研究进展,并用递归序列、同余式、二次剩余的方法证明了丢番图方程x2-Dy4=N在(D,N)=(3,397)时仅有正整数解(20,1)。5.介绍了丢番图方程ax4+bx2y2+cy4=dz2的研究进展,并用Fermat无穷递降法证明了丢番图方程ax4+bx2y2+cy4=dz2当(a,b,c,d)=(2,2,1,1)时无正整数解。更多还原

【Abstract】 Diophantine equation is the integer algebraic equation(or equations) in which the number of variable is more than the number of the equation. Diophantine equation is a very important content and research topic in number theory. It is closely connected with algebraic number theory, combinatorics, algebraic geometry. The achievements in Diophantine equation play an important role both in every branch of mathematics and in other subjects, such as physics, economics, computer science. so there are still many people who have great interested in Diophantine equation.The main contents of this paper are :1. The overview of Diophantine equation, the main achievements of Diophantine equation, The difficulty and principle of solving Diophantine equation are discoursed.2. The preliminary knowledge of the paper are given, including congruence theory,quadratic residue, Legendre symbol, Pell equation Some of the main definitions, nature,theorem.3. The research progress of Diophantine equation Ax2+B=yn is introduced. by using the method of algebraic number theory proved that the Diophantine equation Ax2+B=ynwhen(A,B, n) =(1,4,9) has no integer solution.4. The research progress of Diophantine equation x2-Dy4=N is introduced. by using the method of recurrent sequence, congruence, quadratic remainder proved that the Diophantine equation x2-Dy4=N when(D,N)=(3,397) has only the integer solution(x,y)=(20,1).5. The research progress of Diophantine equation ax4+bx2y2+cy4=dz2 is introduced, and using the method of Fermat’s infinite descent proved that the Diophantine equation ax4+bx2y2+cy4=dz2when(a,b,c,d)=(2,2,1,1) has no positive integer solution.更多还原