【摘要】：不定方程(又稱Diophantine方程)與Smarandache函數(shù)的均值問(wèn)題是數(shù)論中兩個(gè)至關(guān)重要、且較為活躍的數(shù)學(xué)領(lǐng)域,它們有著極其豐富的內(nèi)容,但仍有一些尚未解決的問(wèn)題激發(fā)著許多專家及學(xué)者的研究興趣.本文利用初等方法、解析方法研究了兩類不定方程的可解性問(wèn)題,以及與Smarandache函數(shù)相關(guān)的均值問(wèn)題,主要成果如下:1.利用遞歸數(shù)列,Legendre符號(hào)的性質(zhì),同余的性質(zhì),以及Pell方程的解的性質(zhì)等初等方法討論了不定方程x~3±a~3=Dy~2(D＞0)的整數(shù)解問(wèn)題,分別證明了不定方程x~3+27=37y~2僅有整數(shù)解(x,y)=(-3,0);不定方程x~3-27=37y~2僅有整數(shù)解(x,y)=(3,0),(30,±27),(4,±1);以及不定方程x~2±1331=2pqy~2的整數(shù)解的情況.2.利用初等方法討論了不定方程(na)~x+(nb)~y=(nc)~z的整數(shù)解問(wèn)題,證明了當(dāng)a=20,b=99,c=101時(shí),方程(na)~x+(nb)~y=(nc)~z僅有正整數(shù)解(x,y,z)=(2,2,2).3.利用解析方法研究了Smarandache Ceil函數(shù)與素因子積函數(shù)U(n)的均值分布問(wèn)題,并給出一個(gè)有趣的漸近公式.4.利用初等方法和解析方法研究了Smarandache冪函數(shù)SP(n)的均值問(wèn)題,即在簡(jiǎn)單數(shù)的序列上得到了Smarandache冪函數(shù)SP(n)與數(shù)論函數(shù)R(n)的復(fù)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 鹵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 鹵a~3=Dy~2 (D > 0). Y) = (- 3, 0); There are only integer solutions (x, y) = (3, 0), (30, 鹵27), (4, 鹵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, 鹵27), (4, 鹵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.
【學(xué)位授予單位】：延安大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O156

【參考文獻(xiàn)】

相關(guān)期刊論文 前10條

1 杜先存;;關(guān)于Diophantine方程x~3±1=6pqy~2的整數(shù)解[J];東北師大學(xué)報(bào)(自然科學(xué)版);2015年04期

2 馮蕾;趙西卿;劉建;袁秀芳;;關(guān)于Diophantine方程x~3+1=3pqy~2[J];貴州大學(xué)學(xué)報(bào)(自然科學(xué)版);2015年05期

3 李潤(rùn)琪;;丟番圖方程x~3±27=14y~2的整數(shù)解[J];唐山師范學(xué)院學(xué)報(bào);2015年05期

4 普粉麗;張汝美;楊吉英;;Diophantine方程x~3+5~3=2pqy~2的整數(shù)解[J];湖北民族學(xué)院學(xué)報(bào)(自然科學(xué)版);2015年03期

5 李潤(rùn)琪;;不定方程x~3-125=2pqy~2的整數(shù)解研究[J];湖北民族學(xué)院學(xué)報(bào)(自然科學(xué)版);2015年03期

6 吳成晶;;Smarandache Ceil函數(shù)的均值[J];紡織高�；A(chǔ)科學(xué)學(xué)報(bào);2014年04期

7 陳紅紅;;對(duì)不定方程x~3±27=91y~2整數(shù)解的討論[J];內(nèi)蒙古農(nóng)業(yè)大學(xué)學(xué)報(bào)(自然科學(xué)版);2014年05期

8 杜先存;劉玉鳳;管訓(xùn)貴;;關(guān)于丟番圖方程x~3±5~3=3py~2[J];沈陽(yáng)大學(xué)學(xué)報(bào)(自然科學(xué)版);2014年01期

9 唐剛;;關(guān)于丟番圖方程(45n)~x+(28n)~y=(53n)~z的解[J];西南民族大學(xué)學(xué)報(bào)(自然科學(xué)版);2014年01期

10 廖軍;;關(guān)于丟番圖方x~3-5~3=Dy~2的整數(shù)解的研究[J];西南民族大學(xué)學(xué)報(bào)(自然科學(xué)版);2013年06期

相關(guān)碩士學(xué)位論文 前2條

1 馬婭鋒;關(guān)于Smarandache函數(shù)的幾個(gè)問(wèn)題[D];延安大學(xué);2016年

2 袁泉;一些數(shù)論函數(shù)的推廣及均值問(wèn)題[D];西北大學(xué);2013年

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