Diophantine Problems Related To Figurate Numbers And Polygons

Diophantine equation is the polynomial equation(or equations)in which the number of variables is more than the number of equations,and it is one of the oldest branches of number theory.The problems about the Diophantine equations are called Diophantine problems.UA11 things are numbered,and number is the foundation of all things," geometric symmetry and grace give great charm to the figurate number.From the number to the shape,a polygon is a graph that is described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit in geometry.In this thesis,we consider several Diophantine problems related to the figurate numbers and the polygons.Firstly,we study the Diophantine equations about triangular numbers,and consider the linear combination of two triangular numbers is a perfect square,by the basic properties of Pell’s equation and the theory of congruence,we show that if 2n is not a perfect square,or n=d(d)/2,where d(t)is a simple polynomial,then the Diophantine equation 1+n(y2)=z2,n∈Z+has infinitely many positive integer solutions.For some special integers m,n,we get the Diophantine equation m(x2)+n(y2)=z2,m,n∈Z+has infinitely many positive integer solutions.At the same time,we prove that the Diophan-tine equation z2=a(x2)2+b(x2)(y2)+c(y2)2,a,b,c∈Z has infinitely many positive integer solutions when a,b,c are some special integers.Secondly,we investigate the Heron triangles with figurate number sides and the right angle triangles with sides are values of polynomials.By the theory of Pell’s equation and the method of undetermined coefficients,we show that there exist infinitely many isosceles Heron triangles whose sides are polygonal numbers(except square numbers)and binomial coefficients.Meanwhile,we show that there exist infinitely many right angle triangles whose sides are the values of certain cubic polynomials.Thirdly,we study two polygons with a common area and a common perimeter.By Fermat’s method,we give that there are infinitely many Heron triangle and integral rhombus pairs with a common area and a common perimeter.By computing rational points on hyperelliptic curve,we show that there does not exist any integral isosceles triangle and rhorubus pairs with a common area and a common perimeter.Moreover,by the theory of elliptic curves,we prove that there are infinitely many integral right(resp.isosceles,Heron)triangle and integral right(resp·isosceles)trapezoid pairs with a common area and a common perimeter.At last,we raise some unsolved Diophantine problems.