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

Let { P n } \{P_n\} 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 P_n is a perfect square if and only if n = ± 1 , ± 7 n=\pm 1, \pm 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 x^2-2y^4=-1 are ( x , y ) = ( 1 , 1 ) (x, y)=(1, 1) and ( 239 , 13 ) (239, 13) .