| Diophantine equation is an important subject in number theory. It is greatly connected with coding and computer science etc. The achievements in diophantine equations play significant role both in the development of various branches of mathematics and other non-mathematical disciplines such as physics, biology, geography, finance, etc. Therefore, there are still many people who are interested in diophantine equations.In this paper, we use some methods such as elementary number theory, algebraic number theory, Diophantine approximation theory and obtain the following results:Firstly, using the theorem of Bilu, Hanrot and Voutier about the primitive divisors of Lucas and Lehmer numbers, we discuss the solution of diophantine equation D1x2+D221=yp when D1and D2are some special values.(1) If D2=2,P(?)3(mod8),nis a positive integer,pis an odd prime and D1is an any square free odd prime. The diophantine equation D1x2+D221=yp hasn’t integer solution (x, y),that gcd(x, y)=1.(2) Let D2=2,D1(?)7(mod8),,Pbe an odd prime and nbe a positive integer, the equation D,x2+22n=yp hasn’t integer solution (x, y) which satisfies2ly(3) If D1=p, D2=3, p is a prime, p(?)7(mod8) and n is a positive integer, the equation px2+32n=yp hasn’t integer solution (x, y) which satisfies gcd(x, y)=1.Secondly, we mainly concern the solution of equation Dx2+1=4y5, x,y∈z when D1and D2are some special values.(1) If D=3, then equation has only integer solutions (x,y)=(±1,1).(2) If D=7,11, then equation has no solution.(3) If D=-5, then equation has only integer solutions (x, y)=(±1,-1).Thirdly, let p be an odd primep=3(3k+1)(3k+2)+land kbe a non-negativeinteger, the equation x3+1=3py2has no integer solution. |
