Catalan’s conjecture over number fields

Catalan conjecture/Mihailescu theorem is a theorem in number theory that was conjectured by Mathematician Eugene Charles Catalan in 1844 and was proved completely by Preda Mihailescu in 2005. The note stating the problem was not given the due imprtance at the begining and appeared among errata to papers which had appeared in the earlier volume of Crelle journal. The problem got its due considration after the work of Cassles and Ko Chao in 1960s. The Catalan problem is that the equation x − y = 1 has no solution for x,y,m,n in positive integers with m > 1, n > 1 other than the trivial solution 32 − 23 = 1. An important and first ingredient for the proof is Cassles criteria which says that whenever we have a solution of x − y = 1 with p,q odd primes then q|x and p|y . Here we look a generalization of the problem, namely we will consider the equation xp−yq = 1 where x,y takes value in ring of integers OK of a number field K and p,q are rational primes. In this article we supply a possible formulation of Cassles criterion and a proof for the same in some particular cases of