By explicitly finding the complete set of curves of genus
0
0
or
1
1
in some surfaces of general type, we prove that under the Bombieri-Lang conjecture for surfaces, there exists an absolute bound
M
>
0
M>0
such that there are only finitely many sequences of length
M
M
formed by
k
k
-th rational powers with second differences equal to
2
2
. Moreover, we prove the unconditional analogue of this result for function fields, with
M
M
depending only on the genus of the function field. We also find new examples of Brody-hyperbolic surfaces arising from the previous arithmetic problem. Finally, under the Bombieri-Lang conjecture and the ABC-conjecture for four terms, we prove analogous results for sequences of integer powers with possibly different exponents, in which case some exceptional sequences occur.