On the Periods of Biperiodic Fibonacci and Biperiodic Lucas Numbers

where α = (1 + √5)/2 and β = (1 − √5)/2 are roots of the characteristic equation x2 − x − 1 = 0. The positive root α is known as “golden ratio.” Fibonacci numbers and their generalizations have many interesting properties and applications to almost every field of science and art [1–3]. In particular, Edson and Yayenie [4] introduced the Biperiodic Fibonacci sequence as follows: q0 = 0, q1 = 1, qn = {{ aqn−1 + qn−2, if n is even bqn−1 + qn−2, if n is odd for n ≥ 2, (2)

If we take  =  = 2, we get the Pell sequence {0, 1, 2, 5, 12, 29, 70, . ..}.Similarly, if we take  =  = , we get the -Fibonacci sequence [5][6][7][8].Binet's Formula for the Biperiodic Fibonacci sequence is given as where ,  = ( ± √  2  2 + 4)/2 are the roots of the characteristic equation  2 −  −  = 0 and () =  − 2⌊/2⌋.Moreover, This sequence and its properties can be found in [1,4].Another well-known sequence is the Lucas sequence which satisfies the same recurrence relation as the Fibonacci sequence   =  −1 +  −2 with the initial conditions  0 = 2 and  1 = 1.Binet's Formula for the Lucas sequence is where  and  are defined in (1).Bilgici defined generalization of Lucas sequence similar to the Biperiodic Fibonacci sequence as follows: where  and  are two nonzero real numbers.We take  and  as integers.This sequence and other generalizations of Lucas sequence with their properties can be found in [9,10].
On the other hand several researchers have made significant studies about the period of the recurrence sequences [2].Wall [12] defined the period-length of the recurring series obtained by reducing a Fibonacci series by a modulus .As an example, the Fibonacci sequence mod3 is and has period 8. Vinson [3] and Robinson [13] both extended Wall's study.Moreover, they studied the Fibonacci sequence for prime moduli and showed that for primes  ≡ 1, 4 (mod5) the period-length of the Fibonacci sequence mod divides  − 1, while for primes  ≡ 2, 3 (mod5) the period-length of the Fibonacci sequence mod  divides 2(+ 1).Gupta et al. [14] give alternative proofs of this results that also use the Fibonacci matrix.They place the roots of its characteristic polynomial in an appropriate splitting fields.Renault [15] examined the behaviour of the (, )-Fibonacci sequence under a modulus.
Lucas studied the (, )-Fibonacci sequence extensively.He assigned Δ =  2 + 4 and deduced that if Δ is quadratic residue (that is, a nonzero perfect square) mod , then () |  − 1.If Δ is quadratic nonresidue then () |  + 1.Also, Rogers and Campbell studied the period of the Fibonacci sequence mod [16].They investigated the Fibonacci sequence modulo  prime and then generalized to prime powers.

Period of Biperiodic Fibonacci Sequence
In this section, we investigate the Biperiodic Fibonacci sequence modulo  prime and then generalize to prime powers.
Definition 1.The period of the Biperiodic Fibonacci sequence modulo a positive integer  is the smallest positive integer  such that By the definition above, the only members that can possibly come back to the starting point are multiples of .This can be summed up in the statement that if  is the period of   (mod), then, for any  ∈ . ( Theorem 2. Let  be a prime and let  be a positive integer.If then, We remark that the proof of the Theorem 2 can be seen in [16]. Theorem 3. Let  be a prime, let  be a positive integer, and let  and  be the fundamental roots of the Biperiodic Fibonacci sequence.If  is the period of   (mod), Proof.For  ̸ = 0 and  is a prime integer, we have and then Also we obtain If  is even, then From   ≡   (mod), we get Thus,   ≡   ≡ 1 (mod).From Theorem 2, If  is odd, For   ≡   (mod) and /() (+1)/2 ̸ = 0, Therefore,   ≡   ≡ 1 (mod).From Theorem 2, Theorem 4. Let  be an odd prime, let  denote the period of   (mod), and let  =  2  2 + 4 be a nonzero quadratic residue mod; then  |  − 1.
Theorem 8. Let  be a prime, let  denote the period of   (mod), and let   denote the period of   (mod  ).