On the Exponential Diophantine Equation (a(n)-2)(b(n)-2) = x(2)*

Abstract

In this paper, we deal with the equation (a(n) - 2)(b(n) - 2) = x(2), 2 <= a < b, and a, b, x, n is an element of N. We solve this equation for (a, b) is an element of {(2, 10), (4, 100), (10, 58), (3, 45)}. Moreover, we show that (a(n) - 2)(b(n) - 2) = x(2) has no solution n, x if 2 vertical bar n and gcd(a, b) = 1. We also give a conjecture which says that the equation (2(n) - 2)((2P(k))(n) - 2) = x(2) has only the solution (n, x) = (2, Q(k)), where k > 3 is odd and P-k, Q(k) are the Pell and Pell Lucas numbers, respectively. We also conjecture that if the equation (a(n) - 2)(b(n) - 2) = x(2) has a solution n, x, then n <= 6.