positive integer solution

very basic Diophantine trick is to check that both LHS and RHS should have same parity i.e both should be odd or both should be even

2006^{c}-2004^{a} = 2005^{b}
While LHS will always be even for positive integer a,b,c RHS will always be odd.
Hence no solutions.

:)

Well, (a,b,c)=(0,1,1) is the only solution it seems.

@Sambit
(0,1,1) implies 2005 = 2006!
Is it something like 2=1 (fallacy)

@sambit,
he is asking for positive inegral solutions.

thanks ninepointcircle

sambit for positive Integer solution