The Diophantine Equation $3x^4-2y^2=1$.

Solving a family of Thue equations with an application to the equation x2-Dy4=1

“…We remark that one can show in a similar way that solutions to the equation z 2 = V 4n+3 correspond to approximations x/y which are close to β (1) and β (2) . We do not seem to be able to apply the hypergeometric method in this case, and so the focus of this paper will be entirely on the equation z 2 = V 4n+1 , which is all that we require for the proof of Theorem 1.1.…”

“…We will show that a positive integer solution to z 2 = V 2k+1 (t) with k even gives rise to a solution (x 1 , y 1 ) to (1.4) with x 1 /y 1 close to either β (3) or β (4) . A similar argument shows that a solution to z 2 = V 2k+1 (t) with k odd gives rise to a solution (x 1 , y 1 ) to (1.4) with x 1 /y 1 close to either β (1) or β (2) .…”

“…It is not surprising that Bumby's equation 3X 4 − 2Y 2 = 1 in [2] has the two solutions (1, 1) and (3,11). More generally, if t is an integer of the form t = m 2 + m, with m ≥ 1, then the equation…”

“…The case (a, b) = (2, 1) was solved by Ljunggren in [8], wherein it was shown that (X, Y ) = (1, 1), (13,239) are the only solutions in positive integers. For the case (a, b) = (3, 2), Bumby [2] showed that the only squares in this sequence are v 1 and v 3 . In other words, the only solutions in positive integers X, Y to the equation 3X 4 − 2Y 2 = 1 are (1, 1) and (3,11).…”

“…It is important to note that problems arise when one attempts to apply Thue's method to the entire family of equations (2.1). In particular, there is difficulty in obtaining an effective measure of approximation for the two roots β (1) and β (2) . This is due to the fact that these roots approach √ t as t goes to ∞ (see below), and therefore have no explicit rational approximations suitable for all choices of t. The authors of [3] are able to deal with the particular subclass of equations of the type X 2 − dY 4 = −1 because in this case, the parameter t in (1.4) is a square, in which case one can initiate the hypergeometric method using the rational approximation √ t.…”