Simultaneous Pell equations

Determination of Gaussian Integer Zeroes of F(x,z)=2x4−z3

“…By assuming the Shimura-Taniyama conjecture he also showed that the equation x n + y n = z 3 has no non-trivial primitive solutions for n ≥ 3. More recently, Bakar et al [1] investigated the Diophantine equation 5 x + p m n y = z 2 , where p > 5 is a prime number and y ∈ {1, 2}, presenting the positive solutions under the forms (x, m, n, y, z) = (2r, t, p t k 2 ± 2 k 5 r , 1, p t k ± 5 r ) and (x, m, n, y, z) = 2r, 2t, 5 2r−α −5 α 2p t , 2, 5 2r−α +5 α 2p t , for k, r, t ∈ N and 0 ≤ α < r. Sihabudin et al [10] discussed the system of simultaneous Pell equations x 2 − my 2 = 1 and y 2 − pz 2 = 1, and found the solutions (x, y, z, m) = (y 2 n t ± 1, y n , z n , y 2 n t 2 ± 2t) and (x, y, z, m) =…”