On Y-coordinates of Pell equations which are base 2 rep-digits

Abstract

. In this paper, we show that if ( X k ,Y k ) is the k th solution of the Pell equation X 2 − dY 2 = 1 for some non–square integer d > 1, then the equation Y k = 2 n − 1 has at most two positive integer solutions ( k,n ).