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Wilhelm Ljunggren proved many fundamental theorems on equations of the form aX 2 bY 4 = , where 2 f 1; 2; 4g. Recently, these results have been improved using a number of methods. Remarkably, the equation aX 2 bY 4 = 2 remains elusive, as there have been no results in the literature which are comparable to results proved for the other values of . In this paper we give a sharp estimate for the number of integer solutions in the particular case that a = 1 and b is of a certain form. As a consequence of this result, we give an elementary solution to a Diophantine problem due to Martin Gardner which was previously solved by Charles Grinstead using Baker's theory.

On a family of quatric equations and a Diophantine problem of Martin Gardner