Solutions of the Diophantine Equation x 2 - Dy 4 = k

Abstract

(1) X2 -Dy4 =k appears to be difficult even though it can readily be transformed to the general Pell's equation. For IkI = 1 and for a few small values of D, Eq. (1) has been dealt with in the literature [1]-[4]. For k = 1, Ljunggren [2] proved that Eq. (1) has at most two solutions in positive integers and Mordell [3] found that when D is an odd prime, D =5, 9 or 13 (mod 16) except for D = 5, Eq. (1) has no solutions. In a subsequent paper, Ljunggren [4] showed that the same holds good for all primes p 1 (mod 4). For k = -1, Ljunggren [1] proved that under certain restrictions, Eq. (1) has at most two solutions in positive integers, and he showed that for k = -1 and D = 2, (x, y) = (1, 1) and (239, 13) are the only two solutions and (x, y) = (2, 1) is the only solution for k = -1 and D = 5. For IJkI > 1 and D > 5, Eq. (1) has not been treated so far and the investigations [1], [2] related to finding the upper bound for the number of solutions provide means that are hardly practical for determining these solutions. It was therefore felt desirable to obtain solutions of Eq. (1) by means of a numerical search. The search was conducted for 37 values of D, in the range 2 _ D ? 43, where D is not a perfect square. The parameters y and k were arbitrarily chosen to be y < 2 X 105 and IkI < 999. We anticipate that the results of such an extensive search will be useful for checking some of the conjectures concerning this equation and also provide further insight into the methods of finding complete sets of solutions. Method. Rewrite (1) as