On negative Pell equations: Solvability and unsolvability in integers

Solvability criteria of negative Pell equations x(2) - dy(2) = -1 have previously been established via calculating the length for the period of the simple continued fraction of root d and checking the existence of a primitive Pythagorean triple for d. However, when d >> 1, such criteria usually require a lengthy calculation. In this note, we establish a novel approach to construct integers d such that x(2) - dy(2) = -1 is solvable in integers x and y, where d = d(u(n), u(n+1), m) can be expressed as rational functions of u(n), and u, +1 and fourth-degree polynomials of in, and u n satisfies a recurrence relation: u(0) = u(1) = 1 and u(n+2) = 3u(n)(+1) - u(n), for n is an element of N U {0}. Our main argument is based on a binary quadratic relation between u(n) and u(n+1) and properties 1+u(n)(2)/u(n+1)is an element of N and 1+u(n+1)(2)/ u(n ) is an element of N Due to the recurrence relation of u(n) such d' s are easy to be generated by hand calculation and computational mathematics via a class of explicit formulas. Besides, we consider equation x(2)- k(k 4)m(2)y(2) = -1 and show that it is solvable in integers if and only if k =1 and m is an element of N is a divisor of 1/2 u(3n+2) for some is an element of N U {0}. The main approach for its solvability is the Fermat's method of infinite descent.