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
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.
