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.

ผลลัพธ์ (อังกฤษ) 2:

In this section, we consider Diophantine equations x
2 - Kxy + Y
2 ∓ x = 0 and x
2 - Kxy - Y
2 x = 0. ∓ Discussing Before
we introduce these equations 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, 1 and U1 = Un + 1 = Kun-Un-2 for n ≥ 2, where K ≥ 1, is an integer. Also, U-n = (-1)
n + n ∈ N. 1Un for all
Moreover, the generalized Fibonacci Sequence {un} with parameter K, is defined by u0 = 0, = 1 and u1 = 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 Un Sequence {x} is
2-KX-1 = 0 and the Roots of. this Equation
are alpha =

K +
√
K
2 + 4

/ 2 and alpha = beta =

K -
√
K
2 + 4

/ 2. It is clear that αβ = -1, α2 = kα + 1, and α + β = k. Let
Z [alpha] = {Aα + B: a, B ∈ Z}. Can it then be seen that Z [alpha] is ​​a subring of the Ring of the Real algebraic integer quadratic Field
Q
√
K
2 + 4

and Z [alpha] is ​​Equal to the Ring of the Real algebraic integer quadratic Field Q
. √
K
2 + 4

when K
2 + 4 square is
free.
If Αx Y + Z is in a UNIT [alpha], then Can it be shown that X-
2 + Kxy + Y
2 = (Αx + Y) (Αx. + y) = ± 1. The Proof of the following
theorem is in GIVEN [8]. But we will give its proof for the sake of completeness.

ผลลัพธ์ (อังกฤษ) 3:

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 = (signed 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 signed.√.K.2 + 4../ 2. It is clear that α β = − 1 α, 2 = k α + 1 and, α + β = k. Let.α Z [] = {a α + B: a B ∈, Z}. Then it can be seen that Z [] is α a subring of the algebraic integer ring of the real quadratic. Field.Q. √.K.2 + 4..And Z [] is α equal to the algebraic integer ring of the real quadratic field Q. √.K.2 + 4..When k.2 + 4 is square.Free.If α x+y is a unit in Z [α], then it can be shown that − X.2+kxy+y.2 = (α x + y) (α x + y) = convert 1. The proof of the following.Theorem is given in [8]. But we will give its proof for the sake of completeness.