两类不定方程解的讨论与Smarandache函数均值的研究

发布时间：2019-03-28 19:12

【摘要】：不定方程(又称Diophantine方程)与Smarandache函数的均值问题是数论中两个至关重要、且较为活跃的数学领域,它们有着极其丰富的内容,但仍有一些尚未解决的问题激发着许多专家及学者的研究兴趣.本文利用初等方法、解析方法研究了两类不定方程的可解性问题,以及与Smarandache函数相关的均值问题,主要成果如下:1.利用递归数列,Legendre符号的性质,同余的性质,以及Pell方程的解的性质等初等方法讨论了不定方程x~3±a~3=Dy~2(D＞0)的整数解问题,分别证明了不定方程x~3+27=37y~2仅有整数解(x,y)=(-3,0);不定方程x~3-27=37y~2仅有整数解(x,y)=(3,0),(30,±27),(4,±1);以及不定方程x~2±1331=2pqy~2的整数解的情况.2.利用初等方法讨论了不定方程(na)~x+(nb)~y=(nc)~z的整数解问题,证明了当a=20,b=99,c=101时,方程(na)~x+(nb)~y=(nc)~z仅有正整数解(x,y,z)=(2,2,2).3.利用解析方法研究了Smarandache Ceil函数与素因子积函数U(n)的均值分布问题,并给出一个有趣的渐近公式.4.利用初等方法和解析方法研究了Smarandache幂函数SP(n)的均值问题,即在简单数的序列上得到了Smarandache幂函数SP(n)与数论函数R(n)的复合均值。
[Abstract]:The mean value problem of indefinite equation (also known as Diophantine equation) and Smarandache function is two important and active mathematical fields in number theory, and they are very rich in content. But there are still some unresolved problems that arouse the interest of many experts and scholars. In this paper, the solvability problem of two kinds of indefinite equations and the mean value problem related to Smarandache function are studied by means of elementary method and analytic method. The main results are as follows: 1. By using recursive sequence, the properties of Legendre symbol, congruence and the properties of solution of Pell equation, this paper discusses the problem of integer solution of indeterminate equation x = 3 卤a~3=Dy~2 (D > 0). It is proved that the indeterminate equation x = 3 27=37y~2 has only integer solution (x, n = 3), and that there is only integer solution (x, n = 2) of the equation x = 3 卤a~3=Dy~2 (D > 0). Y) = (- 3, 0); There are only integer solutions (x, y) = (3, 0), (30, 卤27), (4, 卤1) for the indeterminate equation x~3-27=37y~2, and the integer solution for the indeterminate equation x ~ 2 + 1331 = 2pqy~2 is only (x, y) = (3, 0), (30, 卤27), (4, 卤1). In this paper, the integer solution of the indeterminate equation (na) ~ x (nb) ~ y = (nc) ~ z is discussed by means of elementary method. It is proved that the equation (na) ~ x (nb) ~ y = (nc) ~ z only has a positive integer solution (x, y, z) = (2,2,2,2,2,2,2,2,2,2,2,2,2) when a ~ x (nb) ~ y = (nc) ~ z. 2) 3. In this paper, the mean value distribution of Smarandache Ceil function and prime factor product function U (n) is studied by means of analytic method, and an interesting asymptotic formula .4 is given. In this paper, the mean value of Smarandache power function SP (n) is studied by means of elementary method and analytic method, that is, the composite mean of Smarandache power function SP (n) and number theory function R (n) are obtained on the sequence of simple numbers.
【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

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