Some Extensions on Numbers

My previous work dealt finding numbers which relatively prime to factorial value of certain number, high exponents and also find the way for finding mod values on certain number’s exponents. Firstly, I retreat my previous works about Euler’s phi function and some works on Fermat’s little theorem. Next, I construct exponent parallelogram to find coherence numbers of Euler’s phi functioned numbers and apply to Fermat’s little theorem. Then, I test the primality of prime numbers on Pascal’s triangle and explore new ways to construct Pascal’s triangle. Finally, I find the factorial value for certain number by using exponent triangle.

关于整除性n｜φ(n)+σ(n)

本文证明了：ｉ）当合数ｎ至多只有两个不同的素因子时，ｎφ（ｎ）＋σ（ｎ）；ｉ）若奇合数ｎ满足ｎ｜φ（ｎ）＋σ（ｎ），则ｎ至少有６个不同的素因子，且ｎ≥６５１５５１１５０２５；ｉｉ）在区间［１０７，２·１０７］中有且仅有一个ｎ，即ｎ＝１２５５８９１２，满足ｎ｜φ（ｎ）＋σ（ｎ）．

关于欧拉函数方程φ(φ(x))=2t的可解性

对任意的正整数n,函数φ(n)为著名的Euler函数,即在序列1,2,···,n中与n互质的整数的个数.本文利用初等方法研究了方程φ(φ(x))的可解性,并给出了该方程的全部正整数解.

关于方程φ(x)=2t

设t是正奇数.本文给出了方程φ(x)=2t的全部正整数解x,其中φ(x)是Euler函数.

Diophantione方程x^(d(n))+y^(d(n))=z^(φ(n))的本原解

对于正整数n,设d(n)和φ(n)分别是除数的函数和Euler函数,又设p是奇素数.证明了:当n=1,2,4或p时,方程xd(n)+yd(n)=zφ(n)有无穷多组本原解(x,y,z);当n≠1,2,4,p或p2时,该方程无本原解(x,y,z).

An Equivalent Form of Strong Lemoine Conjecture and Several Relevant Results

In this paper, we consider some problems involving Strong Lemoine Conjecture in additive number theory. Based on Dusart's inequality and Rosser-Schoenfeld's inequality, we obtain several new results and give an equivalent form of Strong Lemoine Conjecture.