This Paper is aimed to study a generalization of the Catlan Conjecture (The only nontrivial solution to& x^m-y^n=1& is x=3,m=2,y=2,n=3), that is, to determine the nontrivial solutions to& x^m-2y^n=1&. To solve this problem, I study some arithmetic properties of the solutions to Pell equation, or more precisely, when the minimum solution is fixed, the properties about factors of the recursive solutions to the equation. Furthermore, I apply these properties into the study of& x^m-2y^n=1& and obtain some successful results when n is restricted to be even. What’s more, some applications of these arithmetic properties in other problems are contained in the paper as well.

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In this paper, we prove a good-λ inequality between the nontangential maximal function and the square area integral of a subharmonic function$u$in a bounded NTA domain$D$in$R$^{$n$}. We achieve this by showing that a weighted Riesz measure of$u$is a Carleson measure, with the Carleson norm bounded by a constant independent of$u$. As consequences of the good-λ inequality, we obtain McConnell-Uchiyama's inequality and an analogue of Murai-Uchiyama's inequality for subharmonic functions in$D$.