Study On Integer Solutions Of Some Indefinite Equations

The study of integer solution of Diophantine equation,common solution of Pell equations and the solution of elliptic curve is always a hot topic for number theory enthusiasts,and has been deeply studied by number theory scholars,and also made a lot of research results.This paper mainly uses the typical elementary method in number theory to study the problems related to integer solutions of several kinds of indefinite equations.The main results are as follows:The first part,the integer solution problem Diophantine equation x2+16=32 y17 and x3+1=2247y2 are studied,and get the equation x2+16=32y17 only integer solution(x,y)=(±4,1);The equation x3+1=2247y2 has only trivial integer solutions(x,y)=(-1,0).In the second part,we explore the common solution of Pell equations,and give the common solution of Pell equations x2-56y2=1 and y2-Dz24,x2-(m2+m)y2=1 and y2-Dz2=4.In the third part,the problem of integer points of elliptic curves is studied,and it is proved by classification that the integer point of elliptic curve y2=x(x-181)(x-197)has only trivial points(x,y)=(0,0),(181,0),(197,0);And there are non-tri vial points(x,y)=(49,±42),(423,±7896)as well as trivial points(x,y)=(0,0),(31,0),(47,0)on elliptic curves y2=x(x-31)(x-47).