The Relations Between The Representations Of A Positive Integer As Sums Of Squares And Sums Of Triangular Numbers

The number of representations of a positive integer as sums of squares and sums of triangular numbers is a classical mathematical problem in number theory.Many well-known mathematicians,such as Lagrange and Gauss,have made outstanding contributions to this topic.This topic is closely related to many branches of mathematics and is one of the hot topics in combinatorics and number theory.This thesis focuses on investigating the relationship between the number of representations of n as sums of squares and the number of representations of n as sums of triangular numbers.Let N(a,b,c,d;n)and t(a,b,c,d;n)denote the number of representations of n as ax~2+by~2+cz~2+dw~2 and the number of representations of n as ax~2+by~2+cz~2+dw~2?a(x(x+1))/2+b(y(y+1))/2+c(z(z+1))/2+d(w(w+1))/2,respectively,where a,b,c,d are positive integers,n is a nonnegative integer and x,y,z,w are integers.Recently,Sun established many relations between N(a,b,c,d;n)andt(a,b,c,d;n).At the end of his paper,Sun posed 23 conjectures which state some interesting relationships between N(a,b,c,d;n) and t(a,b,c,d;n).In this thesis,we confirm several conjectures proposed by Sun by establishing Ramanujan's theta function identities.The results of this thesis further strengthen the applications of Ramanujan's theta function in classical subjects of number theory.