| The representation of positive integer is one of the most important topics in number theory and the representations of a positive integer as sums of squares and sums of triangular numbers is one of hot topics in the theory of integer representations.In this thesis,we mainly investigate the relations between sums of squares and sums of triangle numbers.Let T(a1,a2,…,ak;n)denote the number of representations of n as a1x1(x1+1)/2+a2x2(x2+1)/2+…+akxk(xk+1)/2,where a1,a2,…,ak are positive integers,n,x1,x2,…,xk are arbitrary nonnegative integers,and let N(a1,a2,…,ak;n)denote the number of representations of n as a1x12+a2x22+…+akxk2,where this time x1,x2,…,xk are integers.Recently,a number of mathematicians such as Adiga,Baruah,Cooper,Han,Hirschhorn and Sun had discovered some relations between T(a1,a2,…,ak;n)and N(a1,a2,…,ak;n).In particular,Sun not only proved many relationships between T(a1,a2,…,ak;n)and N(a1,a2,…,ak;n),but also posed a lot of conjectures about the relations between them.In this thesis,we confirm some of Sun’s conjectures by using Ramanujan’s theta function identities and(p,k)-parametrization of theta functions.Our main idea can be stated as follows.We first transform some relations between T(a1,a2,…,ak;n)and N(a1,a2,…,ak;n)conjectured by Sun into identities involving theta functions and then prove these identities by employing(p,k)-parametrization and some Ramanujan’s theta function identities. |
