Some Special Classes Of Indeterminate Equation

Diophantine equation is also called indeterminate equation. It is an important branch of number theory, also old and active one in the field of mathematics through the whole history. Diophantine equation not only develops itself very actively, but also extensively applies to use in the research field of other subjects, such as discrete mathematics, economics and physics. It plays an important role in our study, research, and solving the actual problems. Therefore, many researchers have studied the indeterminate equation (equations) extensively and deeply in the domestic and abroad.In this dissertation, several special kinds of Diophantine equation were studied by using elementary methods. The main achievements contained in this paper are as follows:1. Let N+ be the set of positive integer, a,b,c be greater than 1 and each other being co-prime. The equation ax+by=cz, x, y, z∈N+is a kind of fundamen-tal but important exponent Diophantine equation. Famous scholar Terai has brought forward a little conjecture to solution number of above-mentioned indeterminate equation. It attract many expert and scholar to carry out more thorough investigation and discussion on Terai's conjecture from different angles. In this paper, the integer solutions of Diophantine equation ax+by=cz and x2+by=cz are given under some conditions, respectively.2. The integer solutions of the indeterminate equations x3-8=Dy2 (x,y∈N+, gcd(x,y)=1) are discussed when D is an odd prime and D=108s2+1(s being an even number).3. The existence of the solution about the one time indeterminate equation group are discussed and a general form of it under the situation of having a solution is given.