An elementary solution of the Diophantine equation 𝑥²-2𝑦⁴=-1

Let P n \Pₙ\ be the Pell sequence. By combining the congruence properties of recurrence sequences with the law of quadratic reciprocity, it is proved that for odd n n, P n Pₙ is a perfect square if and only if n = ± 1, ± 7 n= 1, 7. This provides an elementary proof for Ljunggren’s result, which asserts that the only positive integer solutions of the Diophantine equation x 2 − 2 y 4 = − 1 x²-2y⁴=-1 are (x, y) = (1, 1) (x, y) = (1, 1) and (239, 13) (239, 13).