Given a positive integer n and the residue class ring Z_(n)=Z/nZ,we set Z_(n)^(x)to be the group of units in Z_(n),i.e.,Z_(n)^(x)={x∈Z_(n):ged(x,n)=1}.Let N_(m)(n)be the number of solutions of x_(1)^(4)+…+x_(m)^(4)...Given a positive integer n and the residue class ring Z_(n)=Z/nZ,we set Z_(n)^(x)to be the group of units in Z_(n),i.e.,Z_(n)^(x)={x∈Z_(n):ged(x,n)=1}.Let N_(m)(n)be the number of solutions of x_(1)^(4)+…+x_(m)^(4)≡0(mod n)with x_(1),…,x_(m)∈Z_(n)^(x).In this note,we determine an explicit expression of N_(m)(n).This extends the results of Sun and Yang in 2014.展开更多
Empirical analysis and comparative research have been made in this essay on the development scale of the cities in our country.With the analysis on scale efficiency,sustainable development capacity and city size distr...Empirical analysis and comparative research have been made in this essay on the development scale of the cities in our country.With the analysis on scale efficiency,sustainable development capacity and city size distribution,we conclude that the most efficient and sustainable is those cities with population from 0.5 million to 2 million.Therefore the most important thing about city development is to improve the efficiency and sustainable development capacity of cities under 0.5 million.Our cities still belong to sub rank size disttibu,hence,consequently,we should develop potential big cities beside towns so as to make every size cities distribute reasonably and promote the urbanization of oue country.展开更多
基金Supported by the Natural Science Foundation of Henan Province(232300420123)the National Natural Science Foundation of China(12026224)。
文摘Given a positive integer n and the residue class ring Z_(n)=Z/nZ,we set Z_(n)^(x)to be the group of units in Z_(n),i.e.,Z_(n)^(x)={x∈Z_(n):ged(x,n)=1}.Let N_(m)(n)be the number of solutions of x_(1)^(4)+…+x_(m)^(4)≡0(mod n)with x_(1),…,x_(m)∈Z_(n)^(x).In this note,we determine an explicit expression of N_(m)(n).This extends the results of Sun and Yang in 2014.
文摘Empirical analysis and comparative research have been made in this essay on the development scale of the cities in our country.With the analysis on scale efficiency,sustainable development capacity and city size distribution,we conclude that the most efficient and sustainable is those cities with population from 0.5 million to 2 million.Therefore the most important thing about city development is to improve the efficiency and sustainable development capacity of cities under 0.5 million.Our cities still belong to sub rank size disttibu,hence,consequently,we should develop potential big cities beside towns so as to make every size cities distribute reasonably and promote the urbanization of oue country.
