Catalan's Conjecture
8 and 9 (
and 
) are the only consecutive Powers (excluding 0 and 1), i.e., the only solution toCatalan's Diophantine Problem. Solutions to this problem (Catalan's Diophantine Problem) are equivalent tosolving the simultaneous Diophantine Equations
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
and 
are Powers, then
(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), andit 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.
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