D(4)-pair and its extension

Abstract

Abstract We prove that if k ⩾ 3 , c and d are positive integers with c d and the set { k − 2 , k + 2 , c , d } has the property that the product of any of its distinct elements increased by 4 is a perfect square, then d is uniquely determined. In the proof we use the standard methods used in solving similar problems. Namely, we firstly transform our problem into solving the system of simultaneous pellian equations which furthermore leads to finding intersection of binary recurrence sequences. We get the lower bound for solutions using mostly the congruence method. Combining it with hypergeometric method in which we give an improvement of known results in our special case we get the desired result for “large” values of parameter k . The remaining values of k are solved using Bakerʼs theory of linear form in logarithms.