Types Of Diophantine Equations

Diophantine equation is an important subject in number theory.It was greatly connected with algebra, combinatorial mathematics and computer science etc.The achievements in diophantine equations play animportant role both in every branch of mathematics and in other subjects,such as physics,economics. So there are still many people who have great interested in diophantine equations.In this paper, the use of the divisible, congruence, as well as elementary number theory, algebraic number theory some of the ways to do the main work as following:Firstly.the solution of diophantine equation D₁x_<sup> + D₂²ⁿ = y^p is given,which applying the theorem of Bilu,Hanrot and Voutier about the primitive divisors of Lucas and Lehmer numbers,received:1. If D₂ = 2,p (?) 3( mod 8),n is a positive integer,p be an odd prime,D be an any square free odd prime ,the diophantine equation Dx² + 2²ⁿ = y^p hasn't integer solution (x,y),that gcd(x,y) = 1.2. Let D (?) 7( mod 8), p be an odd prime,n is a positive integer,the equation Dx² + 2²ⁿ = y^p hasn't integer solution (x,y),that 2|y.3. If D₁ = p,D₂ - 3,pbe a prime,and p (?) 7( mod 8),n > 0 is a positive integer,the equation px² + 3²ⁿ = y^p hasn't integer solution (x, y),that gcd(x, y) = 1.Secondly,we concern the solvability of equation Dx² + 1 = 4y⁵ x,y∈Z(D = 3,7,11,-5)1. If D = 3,only integer solutions (x,y) = (±1,1) of equations ;2. If D = 7,11,then equation has no solution; 3. If D = -5,only integer solutions (x, y) = (±1, -1) of equations .Thirdly.Let p be an odd prime,p = 3(3k + 1)(3k + 2) + 1,k is a non-negative integer,the equation x³ + 1 = 3py² has no integer solution.