Represent Functions And Diophantine Equations

In this thesis,we investigate the representation functions and the Diophantine equations.Now,we introduce our main results.1.Representation functionsLet N be the set of all nonnegative integers.For a set A(?)N,let R2(A,n)denote the number of solutions to n=a+a',a ? A,a' ? A,a<a'.In 2011,Chen studied,in the paper published in Sci.China Math.,the distribution of the values of R2(A,n).In this thesis,we improve some results of Chen.One of our results is as follows:Let N be a positive integer and let A be a subset of N.If R2(A,n)=R2(N\A,n)for all integers n?2N-1,then there exists a positive real number ? such that2.Diophantine equationsLet a? 2,b? 1 be integers and let the square-free part of b be 2pq with p,q two distinct odd primes.In this thesis,we prove that the system of equations has solutions in positive integers if and only if 8a2(2a2-1)/b or 8a2(2a21)(4a2-3)(4a2-1)/b is a perfect square.When 8a2(2a2-1)/b is a perfect square,the solution is when 8a2(2a2-1)(4a2-3)(4a2-1)/b is a perfect square,the solution is This improves a result of Cipu published in Proc.Amer.Math.Soc.in 2018.