Number Theory 725Combining these, we see that we must have eitherx=1ory=1. Either of these
implies the other and gives the solution(v+ 1 ,v, 1 , 1 ).
Remark.Catalan conjectured in 1844 a more general fact, namely that the Diophantine
equationux−vy=1 subject to the conditionx, y≥2 has the unique solution 3^2 − 23 =1.
This would mean that 8 and 9 are the only consecutive powers. Catalan’s conjecture was
proved by P. Mihailescu in 2002. ̆
(Kvant(Quantum), first solution by R. Barton, second solution by R. Stong)