一个数论函数方程x^y-[x]~y=y

The Arithmetic Functional Equation x^y-[x]~y=y

利用初等数论和极限理论研究了一个包含Gauss取整函数方程xy-[x]y=y的可解性问题,证明了当x∈[n,n+1),n∈瓔时,有且只有一个y与之对应,从而方程xy-[x]y=y有无穷多组实数解.同时,当y值非常小时,可以获得解x的具体形式,即在y=2,3时,给出了对应解x的具体形式.

Using the elementary methods and the theory of limits,the solvability of the function equation involving the Gauss function xy-[x]y=y is researched,it proved that the equation has only one y when x∈[n,n＋1）,n∈N.Thus,the equation has infinite solutions.If y is small enough,we can obtain the exact solution x.When y is fixed as 2,3,we gave the proper x correspondingly.