Several Diophantine Equations Related To Polygonal Numbers And Triangles

Diophantine equation is an ancient branch of number theory,the research object is the polynomial equation or system with the number of variables is more than the number of equations,and the solutions are rational or integer numbers.Many problems about the polygonal numbers and the Heron triangles can be transformed into the problems of solving Diophantine equations.In this thesis,we mainly consider the Diophantine equations about the polygonal numbers and the Heron triangles.Firstly,we discuss the problem that the linear combination of two polygonal numbers is a perfect square.By the theory of Pell’s equation and congruence,we study that the linear sum and difference of two polygonal numbers are perfect squares.More precisely,we give the sufficient conditions for the existence of positive integer solutions of the following five Diophantine equations 1+nPk(y)=z2,mPk(x)+nPk(y)=z2,mPk(x)-1=z2,mPk(x)-nPk(y)=z2,and mPk(x)-nPk(y)=1,where(?),x≥1,k≥3 denotes the x-th k-gonal number.Secondly,we consider the problem that the sides of Heron triangles are the values of some polynomials of the same type or different types.By the method of undetermined coefficients and the theory of Pell’s equation,we obtain infinitely many isosceles Heron triangles whose sides are the values of some special polynomials with degree 2 or n(≥ 3).At last,we raise some related problems for further study.