Catalan's Conjecture

8 and 9 ($2^3$ and $3^2$) are the only consecutive Powers (excluding 0 and 1), i.e., the only solution to Catalan's Diophantine Problem. Solutions to this problem (Catalan's Diophantine Problem) are equivalent to solving the simultaneous Diophantine Equations

$$\begin{eqnarray*} X^2-Y^3&=&1\\ X^3-Y^2&=&1. \end{eqnarray*}$$

This Conjecture has not yet been proved or refuted, although it has been shown to be decidable in a Finite (but more than astronomical) number of steps. In particular, if $n$ and $n+1$ are Powers, then $n<\mathop{\rm exp}\nolimits \mathop{\rm exp}\nolimits \mathop{\rm exp}\nolimits \mathop{\rm exp}\nolimits 730$ (Guy 1994, p. 155), which follows from R. Tijdeman's proof that there can be only a Finite number of exceptions should the Conjecture not hold.

Hyyro and Makowski proved that there do not exist three consecutive Powers (Ribenboim 1996), and it is also known that 8 and 9 are the only consecutive Cubic and Square Numbers (in either order).

See also Catalan's Diophantine Problem

References

Guy, R. K. ``Difference of Two Power.'' §D9 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 155-157, 1994.

Ribenboim, P. Catalan's Conjecture. Boston, MA: Academic Press, 1994.

Ribenboim, P. ``Catalan's Conjecture.'' Amer. Math. Monthly 103, 529-538, 1996.

Ribenboim, P. ``Consecutive Powers.'' Expositiones Mathematicae 2, 193-221, 1984.