Catalan's Diophantine Problem

Find consecutive Powers, i.e., solutions to

$$a^b-c^d=1,$$

excluding 0 and 1. Catalan's Conjecture is that the only solution is $3^2-2^3=1$, so 8 and 9 ($2^3$ and $3^2$) are the only consecutive Powers (again excluding 0 and 1).

See also Catalan's Conjecture

References

Cassels, J. W. S. ``On the Equation $a^x-b^y=1$. II.'' Proc. Cambridge Phil. Soc. 56, 97-103, 1960.

Inkeri, K. ``On Catalan's Problem.'' Acta Arith. 9, 285-290, 1964.