In this section, we consider Diopha

In this section, we consider Diophantine equations x
2 − kxy + y
2 ∓ x = 0 and x
2 − kxy − y
2 ∓ x = 0. Before discussing
these equations we introduce two kinds of generalized Fibonacci sequences {Un} and {un}. For more information about
generalized Fibonacci sequences one can consult [5–7]. The generalized Fibonacci sequence {Un} with parameter k, is defined
by U0 = 0, U1 = 1 and Un = kUn−1 + Un−2 for n ≥ 2, where k ≥ 1, is an integer. Also, U−n = (−1)
n+1Un for all n ∈ N.
Moreover, the generalized Fibonacci sequence {un} with parameter k, is defined by u0 = 0, u1 = 1 and un = kun−1 − un−2
for n ≥ 2 where k ≥ 3, is an integer. Also u−n = −un for all n ∈ N. When k = 1, we get Un = Fn.
The characteristic equation of the recurrence relation of the sequence {Un}is x
2−kx−1 = 0 and the roots of this equation
are α =

k +
√
k
2 + 4

/2 and α = β =

k −
√
k
2 + 4

/2. It is clear that αβ = −1, α2 = kα + 1, and α + β = k. Let
Z [α] = {aα + b : a, b ∈ Z}. Then it can be seen that Z [α] is a subring of the algebraic integer ring of the real quadratic field
Q
√
k
2 + 4

and Z [α] is equal to the algebraic integer ring of the real quadratic field Q
√
k
2 + 4

when k
2 + 4 is square
free.
If αx+y is a unit in Z [α], then it can be shown that −x
2+kxy+y
2 = (αx + y) (αx + y) = ±1. The proof of the following
theorem is given in [8]. But we will give its proof for the sake of completeness.