On The Study Of The Various Kinds Of Indeterminate Equation

Indeterminate equation is one of the oldest branch of number theory, Diophantine,who comes from ancient Greek, began the study of indeterminate equation as early as the3rd century, so we often called indeterminate equation as indeteminate equation. Indeterminate equation is the equation that the number of variable in the equation is more than the number of equation, it is an important subject in number theory. With the rapid development of the various disciplines in recent years, indeterminate equation has made an unprecedented progress, it not only develops itself actively, but also has a great help for other disciplines (such as physics, economics, biology), therefore it has an active impact on people’s production, life and work to study the related problems of the indeterminate equation, due to the importance of the indeterminate equation, it becomes the focus of many scholars to work on. There are three problems to solve about the study of the indeterminate equation:①The condition it needed when the equation has solution.②The number of the solutions it has when the equation has solution.③Getting all the solutions of the equation.In this dissertation, two kinds of indeterminate equation and one indeterminate equations were studied by using the theory of congruence and the elementary. The main achievements contained in this paper are as follows:1. By using elementary method, this paper proves the indeterminate equation x(x+1)(x+2)=2p2y3has no positive integer solution, where p is even prime.2. Let N+be the set of positive integer, a,b,c be greater than1and each other being co-prime. The equation ax+by=cz,x,y,z∈N+is a kind of fundamental but important exponent indeterminate equation. In1994, famous scholar Terai has brought forward some conjectures to the indeterminate equation it attract many expert and scholar to carry out more thorough investigation and discussion on Terai’s conjecture.This paper proves the indeteminate equation ax+by=cz has only one positive integer solution (x,y,z)=(2,2,3) by using the theory of Gel’fond-Baker, let α=m3-3m,b=3m2-1,c=m2+1,when m is an even integer,and m≥246.3.In this paper,it proves that if where pi and qj are diverse odd primes, pi三5(mod8), qj三7(mod8) and ι≤3, the simultaneous equations x2-6y2=1and y2-Dz2=4only have the integer solution z=140.