Several Diophantine Equations Concerning The Area,Perimeter And Sides Of Triangles

Diophantine equation is the polynomial equation(or system of polynomial 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.In plane geometry,the triangle is a closed figure composed of three line segments which are not on the same line in the same plane.The study of the triangle is usually related to the Diophantine equations in number theory.There are many kinds of triangles,in this thesis,we consider several Diophantine equations concerning the area,perimeter and sides of right triangle,Heron triangle and rational triangle.Firstly,we give a complete answer to the problem of Fermat,and generalize it to the case of Heron triangle.By using the theory of elliptic curve,we get all rational right triangles such that the sum of the area and the square of the sum of two legs is a perfect square,and prove that there are infinitely many Heron triangles such that the sum of the area and the square of the sum of two sides is a perfect square.For the problem of Frenicle,we give partial answers and generalize it.By using the basic properties of Pell’s equation and the theory of congruence,we show that there are infinitely many non-primitive right triangles such that the sum of the area and the hypotenuse(the smaller leg)is a perfect square.For the case of primitive right triangles,by using the properties of Fibonacci sequence,we prove that there exist infinite number of primitive right triangles such that the sum of the area and the smaller leg(the larger leg)is a perfect square.By using the theory of elliptic curve,we show that there are infinitely many rational triangles such that the sum of the area and the largest side(the smallest side)is a perfect square.Secondly,we consider the Diophantine equations about the area and the perimeter of right triangles(Heron triangles).By using the theory of elliptic curve,we obtain all rational right triangles such that the sum of the area and the square of semi-perimeter is a perfect square.Meanwhile,we get two results:1)if n=k2,m>(?),or m=6k2-24n,k>8,0<n<1/4k2,then there is a right triangle such that the sum of m times the area and n times the square of perimeter is a perfect square;2)ⅰ)when n=k2,0<n<3k2-(?),or m>3k2+(?),ⅱ)when m=6k2-2n,0<n<3/2k2-(?),or 3/2k2+(?)<n<3k2,there exists a right triangle such that the sum of m times the area and n times the perimeter is a perfect square.Furthermore,we extend these problems to rational triangles and obtain two results with conditions:1)if P4=(X,Y)is a rational point of infinite order on the elliptic curveε4:Y2=X3-m2v3(v+1)2X+4m2nv4(v+1)4 and satisfies the condition 0<|X|<m(v+1)v3/2,then there are infinite number of rational triangles such that the sum of m times the area and n times the square of perimeter is a perfect square;2)if P5=(X,Y)is a rational point of infinite order on the elliptic curveεs:Y2=X3-27(v+1)2(3m2v3+4n2)X+54n(v+1)3(9m2v3+8n2)and satisfies the condition |X-6n(v+1)|>9m(v+1)v3/2,then there are infinitely many rational triangles such that the sum of m times the area and n times the perimeter is a perfect square.Thirdly,we generalize the result of MacLeod.By using the theory of elliptic curve,we prove that there is no right triangle such that the product of the perimeter and the hypotenuse is a perfect square,and obtain all right triangles such that the product of the semi-perimeter and the hypotenuse is a perfect square.For the case of Heron triangles,by using the method of parameterization,we show that there are infinitely many Heron triangles such that the perimeter(semi-perimeter)and the largest side are both perfect squares,and there exist infinite number of Heron triangles such that the product of the perimeter(semiperimeter)and the largest side is a perfect square.At last,we raise some related Diophantine equations.