In this paper,we firstly modified the mistake in reference of Bennett,then using Strmer s theorem of the solutions of Pell equation,and a deep result of privitive divisor of Bilu,Hanrot and Voutier,we proved that there is no exist four distinct triangular numbers in geometric progression,therefore we sovled the question of Sierpinski on triangular numbers.

By using the result of Lucas number primitive divisor,as n>max{30,12(|b|+1)},it receives all f(x) of irreducible quadratic factors with integer coefficients when the first coefficient equals 1.

By applying the deep theorem of Bilu,Hanrot and Voutier about the existence of primitive divisors of Lucas numbers,we prove that the exponential diophantine equation x2+3m=yn has only positive integer solution(x,y,m,n)=(46,13,4,3) with n>2 and gcd=(x,y)=1.

This thesis is to study the integer solutions and the number of theinteger solutions of some exponential diophantine equations by applyingthe deep theorem of Bilu, Hanrot and Voutier about the existence ofprimitive divisors of Lucas and Lehmer numbers, some fine results on therepresentation of the solutions of quadratic Diophantine equations and theclass number of quadratic field.

We apply a deep result of Bilu,Hanrot and Voutier on primitive divisors to show that if the class number of quadratic field Q((-b)~(1/2))is a power of 2,then,the Diophantine equation x~2 + b~y=c~z has only the positive integes solution (x,y,z)=(a,2,5)with min(x,y,z)>1.