Smarandache函数的几类相关方程的解
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摘要
设φ(n),S(n)分别表示正整数n的Euler函数和Smarandache函数,利用初等的方法和技巧,依据Smarandache函数计算公式,给出k的方程φ(p~αm)=S(p~(ακ))的所有解,其中p为素数,α,m为正整数且gcd(m,p)=1,由此得到方程φ(n)=S(n~k)的所有解(n,k)进而确定了满足条件S(n)|σ(n)的全部正整数n.最后,根据莫比乌斯变换反演定理证明了方程φ(n)=∑_(d|n)S(d)仅有两个解,分别为n=2~5和n=3×2~5.
Let φ(n), S(n) be the Euler function and the Smarandache function of the positive integer n, respectively. Based on elementary methods and techniques, according to the algorithm formula of the Smarandache function, all solutions of the equationφ(p~αm) = S(p~(ακ)) are given, where p is a prime, a and m are both positive integers,and gcd(m,p) = 1. And then we get the solutions of the equation φ(n) = S(n~k). Furthermore, all positive integers n satisfying the condition S(n)|σ(n) are determined.At last, basing on the Mobius transformation inversion theorem, we prove that the equation φ(n) =∑_(d|n)S(d) has only two solutions,namely,n= 2~5 and n = 3 × 2~5.
Let φ(n), S(n) be the Euler function and the Smarandache function of the positive integer n, respectively. Based on elementary methods and techniques, according to the algorithm formula of the Smarandache function, all solutions of the equationφ(p~αm) = S(p~(ακ)) are given, where p is a prime, a and m are both positive integers,and gcd(m,p) = 1. And then we get the solutions of the equation φ(n) = S(n~k). Furthermore, all positive integers n satisfying the condition S(n)|σ(n) are determined.At last, basing on the Mobius transformation inversion theorem, we prove that the equation φ(n) =∑_(d|n)S(d) has only two solutions,namely,n= 2~5 and n = 3 × 2~5.
引文
[1]Bai H. R., Liao Q. Y., Some generalizations of Smarandache function, J. Sichuan Normal Univ., 2018, 41(1):32-38.
[2]Chen B., An equation involving Smarandache function and the Euler function, J. Southwest Univ., 2012,34:70-73.
[3]Farris M., Mitshell P., Bounding the Smarandache function, Smarandache Notions Journal, 2002, 13:37-42.
[4]Gorski D., The Pseudo-smarandache Function, Smarandache Notions Journal, 2002, 13(1-3):140-149.
[5]Kashihara K., Comments and Topics on Smarandache Notions and Problems, New Mexica Erhus University Press,, 1996.
[6]Le M. H., A lower bound for(2~(P-1)(2~P-1)), Smarandache Notions Journal, 2001, 12(1):217-218.
[7]Liao Q. Y., Luo W. L., The explicit formula for the Smarandache function and solutions of the related equations, J. Sichuan Normal Univ., 2017, 40:1-10.
[8]Liu Y. M., On the solutions of an equation involving the Smarandache function, Scientia Magna, 2006, 2(1):76-79.
[9]Smarandache F., Only Problems, Not Solution, Xiquan Publishing House, Chicago, 1993.
[10]Wen T. D., A lower bound estimate of the Smarandache function(in Chinese), Pure and Applied Mathematics, 2010, 26(3):413-416.
[11]Xu Z. F., the value distribution of Smarandache function(in Chinese), Acta Mathematica Sinaca, 2006,49(5):1009-1012.
[12]Xing W. Y., On the Smarandache function, Research on Smarandache Problem in Number Theory, 2005, 2:103-106.
[13]Yi Y., An equation involving the Euler function and Smarandache function, Scientia Magna, 2005, 1(2):172-175.
[14]Zhang W. P., On two problems of the Smarandache function, Journal of Northwest University, 2008, 38(2):173-176.
[2]Chen B., An equation involving Smarandache function and the Euler function, J. Southwest Univ., 2012,34:70-73.
[3]Farris M., Mitshell P., Bounding the Smarandache function, Smarandache Notions Journal, 2002, 13:37-42.
[4]Gorski D., The Pseudo-smarandache Function, Smarandache Notions Journal, 2002, 13(1-3):140-149.
[5]Kashihara K., Comments and Topics on Smarandache Notions and Problems, New Mexica Erhus University Press,, 1996.
[6]Le M. H., A lower bound for(2~(P-1)(2~P-1)), Smarandache Notions Journal, 2001, 12(1):217-218.
[7]Liao Q. Y., Luo W. L., The explicit formula for the Smarandache function and solutions of the related equations, J. Sichuan Normal Univ., 2017, 40:1-10.
[8]Liu Y. M., On the solutions of an equation involving the Smarandache function, Scientia Magna, 2006, 2(1):76-79.
[9]Smarandache F., Only Problems, Not Solution, Xiquan Publishing House, Chicago, 1993.
[10]Wen T. D., A lower bound estimate of the Smarandache function(in Chinese), Pure and Applied Mathematics, 2010, 26(3):413-416.
[11]Xu Z. F., the value distribution of Smarandache function(in Chinese), Acta Mathematica Sinaca, 2006,49(5):1009-1012.
[12]Xing W. Y., On the Smarandache function, Research on Smarandache Problem in Number Theory, 2005, 2:103-106.
[13]Yi Y., An equation involving the Euler function and Smarandache function, Scientia Magna, 2005, 1(2):172-175.
[14]Zhang W. P., On two problems of the Smarandache function, Journal of Northwest University, 2008, 38(2):173-176.