Three Diophantine equations concerning the polygonal numbers

Many authors investigated the problem about the linear combination of two polygonal numbers being a perfect square, i.e., the Diophantine equation mPₖ(x)+nPₖ(y)=z², where Pₖ(x) denotes the x-th k-polygonal number and m, n are positive integers. In this note, we continue the study of this problem in another direction and consider three Diophantine equations mPₖ(x)-1=z², mPₖ(x)-nPₖ(y)=z², mPₖ(x)-nPₖ(y)=1. By the theory of Pell equations and congruences, we obtain some conditions such that the above three Diophantine equations have infinitely many positive integer solutions.