A Research On The Integer Solutions Of Indefinite Equations And Integral Point Solutions Of Elliptic Curves

Indefinite equations and elliptic curves are important fields in number theory,and the research on the integer solutions to indefinite equations and elliptic curves is also regarded as a crucial topic of number theory.The study of indefinite equations can be traced back to the period of Diophantine in ancient Greece,and it still attracts the interest of the number theory workers till now.The theories,methods and results obtained in the study of indefinite equations are of great value,which are the basis for further study of complex problems.The arithmetic theory of elliptic curve was only produced and developed in the 20th century.It is an important branch of modern number theory and has a very close connection with algebraic geometry,an extremely important branch of modern mathematics.Andrew Wiles is a top mathematician on elliptic curves,and he proved Fermat's Last Theorem in the process of effective use of elliptic curve theory,so that elliptic curve theory immediately aroused high attention and enthusiasm for research,and it has been continued to this day.In this thesis,the above-mentioned two parts are discussed by learning the relevant basic knowledge of number theory and reading a large number of literature.The main research results are as follows:In the first part,the four problems of integer solutions of indefinite equations such as x~3±8=Dy~2 and x~3±1=Dy~2 are discussed,and all the integer solutions of them are obtained.Specifically,for x~3+8=19y~2,there are only integer solutions(x,y)=(-2,0),(62,±112);for x~3-8=13y~2,there are infinitely many sets of integer solutions:when x is even,there are only solutions(x,y)=(2,0),(6,±4),(626,±4344),and when it is odd,the concrete formula of integer solutions is given;for x~3-1=4781y~2 and x~3+1=5789y~2,there are only integer solutions(x,y)=(1,0)and(x,y)=(-1,0)respectively.In the second part,the four types of elliptic curves such as y~2=x~3+ax+b are discussed.For them,either the number of integer solutions is estimated,or all the integer solutions are obtained.In detail,for y~2=nx(x2+256),there are at most two integer points except(x,y)=(0,0);for y~2=x~3+63x±134,there are only integer points(x,y)=(-2,0)and(x,y)=(-2,0)respectively;for y~2=x~3+75x-158,there is only integer point(x,y)=(2,0);for y~2=x~3+325x-658,there are only integer points(x,y)=(2,0),(38,±258).