This does not get you very far, but the numbers that you are looking at grow very quickly. So, once small coincidences are ruled out, chance alone mitigates against any equalities or even close calls. It might be possible that in the set of all $X(k,i)$ and $Y(k,j)$ ($i,j \ge 2)$ the smallest distance is $16$ and that only occurs a few times. We now [know for sure][1] that the set of powers $a^i$ (starting at $16$) and the set of near  powers $b^j\pm1$ ($i,j \ge 2$) are disjoint. Your sets are sparser than these. The first two members for your series are $a^2+a-1$ and $a^3+a^2-2a-2$ for $a=4k-2$ along with $b^2-b-1$ and $b^3-b^2-2b+1$ for  $b=4k+2.$ To show the quadratic members are distinct is easy. To have a quadratic and a cubic equal would mean having an integer point on an eliptic curve, so that should be resolvable. It might be that not too much more work is needed to discount the case of equalities involving a mix of second,third,fourth and perhaps even sixth powers. If all that is ruled out then your sets are particularly sparse. 


  [1]: http://en.wikipedia.org/wiki/Catalan's_conjecture