两类Diophantine方程的求解和一类Smarandache函数方程的研究

发布时间：2018-04-28 03:59

本文选题：Diophantine方程 + Euclid整环　；参考：《延安大学》2017年硕士论文

【摘要】：Diophantine方程与包含Smarandache函数的方程问题是数论中的两个热门研究课题,它们的研究成果极大地丰富了数论的理论内容,同时又对其他学科起着至关重要的作用.因此,有必要继续对这两类方程进行深入研究.本文利用初等、代数方法就数论中的两类Diophantine方程和一类Smarandache函数方程整数解的问题进行研究.主要研究成果如下:1.讨论了Diophantine方程x~2+D=4y~5在二次Euclid整环(?)中整数解的问题.首先证明当D=3时,该方程在虚二次Euclid整环(?)中仅有整数解(x,y)=(±1,1);其次分别证明当D=-3,-6,-7,-37时,在其所对应的实二次Euclid整环(?)中均无整数解.2.讨论了Diophantine方程x~2+4~k=y~9在Gauss整环中的可解性问题,证明当k=4时,该方程仅有整数解(x,y)=(±16,2),而当k=3,5时均无整数解.3.讨论了一类包含Smarandache函数S(n)和广义欧拉函数φ_2(n)的方程φ_2(n)=S(n~k),7,8,9的可解性问题,结合C++、python程序,最终得到了该类方程的所有正整数解。
[Abstract]:Diophantine equation and equation problem including Smarandache function are two hot research topics in number theory. Their research results greatly enrich the theoretical content of number theory and play an important role in other disciplines. Therefore, it is necessary to further study these two kinds of equations. In this paper, we use elementary and algebraic methods to study the integer solutions of two classes of Diophantine equations and a class of Smarandache function equations in number theory. The main research results are as follows: 1. The Diophantine equation x2 D=4y~5 in quadratic Euclid domain is discussed. The problem of integer solution in. First of all, it is proved that when D = 3, the equation is in the imaginary quadratic Euclid domain. There is only one integer solution (卤1 ~ 1) and then it is proved that the corresponding real quadratic Euclid domain can be obtained when DX ~ (-3) ~ (-3) -6 ~ (-7) -7 ~ (-37). There is no integer solution. In this paper, the solvability of Diophantine equation XN 2 4~k=y~9 in Gauss domain is discussed. It is proved that when k = 4, the equation has only an integer solution (卤16 ~ 2 ~ 2), but when KG is 3, 5, there is no integer solution. In this paper, we discuss the solvability of a class of equations containing Smarandache function (n) and generalized Euler function (蠁 s _ 2 n). Finally, we obtain all positive integer solutions of this kind of equation by using C ~ (python) program.
【学位授予单位】：延安大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O156

【参考文献】