On bi-periodic Padovan and Perrin quaternions over finite fields

ON BI-PERIODIC P ADO V AN AND PERRIN QUA TERNIONS O VER FINITE FIELDS DIANA SA VIN AND ELIF T AN Abstract. In this paper, w e in vestigate bi-p erio dic Pado v an and bi-p erio dic Perrin quaternions o ver the quaternion algebra Q Z p . W e in tro duce the bi-p erio dic Perrin se- quence an d clarify its structural relationship with the bi-p erio dic P adov an sequence. By extending these sequences to the quaternion setting, we analyze their norm prop erties in the modular framework. F or suitable c hoices of t win prime co efficien ts, we deriv e explicit criteria c haracterizing zero divisors and inv ertible elements in Q Z p . 1. Intr oduction The Fib onacci sequence, denoted b y { F n } n ≥ 0 , is defined b y the second-order recur- rence relation F n = F n − 1 + F n − 2 , n ≥ 2, with the initial conditions F 0 = 0 and F 1 = 1. This simple y et p o w erful recursive definition provides a fundamental mo del for describing v arious natural phenomena associated with gro wth and developmen t. The study of gen- eralizations of the Fib onacci sequence and higher-order recurrence relations has attracted considerable atten tion in the literature since they found profound connections across a wide range of disciplines, including mathematics, computer science, ph ysics, c hemistry , biology , and ev en so ciology . One of the most notable third-order in teger sequences in this con text is the P ado v an sequence [13]. The P ado v an sequence { p n } n ≥ 0 , listed as A000931 in the On-Line Encyclop edia of In teger Sequences [12], is defined b y p n = p n − 2 + p n − 3 , n ≥ 3, with the initial terms p 0 = 1 , p 1 = 0 , and p 2 = 1. The first few terms of the sequence are: 1 , 0 , 1 , 1 , 1 , 2 , 2 , 3 , 4 , 5 , 7 , 9 , 12 , 16 , 21 , 28 , 37 , 49 , 65 , 86 , 114 , . . . . The P errin sequence { r n } n ≥ 0 , listed as sequence A001608 in [12], satisfies the same re- currence relation as Pado v an sequence but b egins with the initial v alues r 0 = 3 , r 1 = 0 , and r 2 = 2 . The first few terms of the sequence are: 3 , 0 , 2 , 3 , 2 , 5 , 5 , 7 , 10 , 12 , 17 , 22 , 29 , 39 , 51 , 68 , 90 , 119 , 158 , 209 , 277 , . . . . 2000 Mathematics Subje ct Classific ation. 11A07, 11B37, 11B39, 11R52, 16G30. Key wor ds and phr ases. Congruences, quaternions, Pado v an num b ers, Perrin num bers, zero divisor, finite fields, t win primes. 1 2 DIANA SA VIN AND ELIF T AN These t wo sequences are connected b y the well-kno wn relation r n = 3 p n − 3 + 2 p n − 2 , n ≥ 3 . A detailed exp osition of the Fib onacci sequence and related integer sequences, we refer to the b o oks [8, 9]. On the other hand, quaternions, originally introduced b y Hamilton, can b e viewed as an extension of complex num bers and hav e found applications in v arious fields such as computer science, ph ysics, differen tial geometry , and quan tum mec hanics. Let F be a field with characteristic not 2 . The generalized quaternion algebra ov er a field F is defined as: Q F ( s, t ) =  x + yi + z j + w k | x, y , z , w ∈ F , i 2 = s, j 2 = t, ij = − j i = k  (1.1) where s and t are nonzero elements of F . It is kno wn that the algebra Q R ( − 1 , − 1) is the classical real quaternion algebra. A generalized quaternion algebra is a division algebra if and only if a quaternion with a norm of zero is necessarily the zero quaternion. In other w ords, for X ∈ Q F ( s, t ) , the norm of X, denoted as N ( X ) and defined as N ( X ) = x 2 − sy 2 − tz 2 + stw 2 equals zero if and only if X = 0. Otherwise, the algebra is called a split algebra. In particular, while the real quaternion algebra is a division algebra, the quaternion algebra o v er the finite field Z p , denoted by Q Z p ( − 1 , − 1) and abbreviated as Q Z p , is kno wn to be a split algebra for any o dd prime p ; see [2]. Sp ecial elements in quaternion algebras ov er finite fields hav e also b een studied in [11, 15, 16, 18 – 20], to whic h w e refer for further bac kground. In recen t y ears, there has been gro wing in terest in quaternion sequences defined via spe- cial integer sequences suc h as the Fibonacci, Lucas, Pell, Pado v an, and P errin sequences. The algebraic prop erties of these quaternion sequences ov er v arious quaternion algebras ha v e been in v estigated b y man y authors. In particular, Horadam [4] defined the Fib onacci quaternions o v er the real quaternion algebra as Q n = F n + F n +1 i + F n +2 j + F n +3 k , where F n denotes the n -th Fib onacci num b er. P ado v an quaternions were later in tro duced b y T a¸ scı [27]. F urther studies on quaternions asso ciated with sp ecial integer sequences can b e found in [4 – 7, 10, 22 – 25]. F or quaternion sequences defined ov er finite fields, we refer the reader to [21, 26]. In this pap er, w e consider the bi-p erio dic Pado v an sequence { P n } n ≥ 0 , in tro duced by Disk ay a and Menk en [1], which is defined b y the following recurrence relation: P n = ( aP n − 2 + P n − 3 , if n is ev en , bP n − 2 + P n − 3 , if n is o dd , for n ≥ 3 , with the initial v alues P 0 = 1 , P 1 = 0 , and P 2 = a . F rom [1], the bi-p erio dic P ado v an n um b ers satisfy the following recurrence relation P n = ( a + b ) P n − 2 − ab P n − 4 + P n − 6 , n ≥ 6 (1.2) ON BI-PERIODIC P ADO V AN AND PERRIN QUA TERNIONS 3 and the generating function of the bi-p erio dic P adov an sequence is giv en by G ( x ) = 1 − bx 2 + x 3 1 − ( a + b ) x 2 + abx 4 − x 6 . (1.3) Recen tly , Gungor et al. [3] provided a combinatorial in terpretation of these n um b ers using the w eigh ted tiling approach. While bi-p erio dic Pado v an n um b ers hav e already b een studied in the literature, an analogous bi-p erio dic extension of the Perrin sequence has not y et b een systematically in v estigated. One of the main ob jectives of this pap er is to introduce the bi-p erio dic Per- rin se quenc e and to establish its fundamen tal relationship with the bi-p erio dic P ado v an sequence. F urthermore, another key motiv ation of this study is to consider the quater- nion sequences asso ciated with these recurrences, namely the bi-p erio dic Pado v an and bi-p erio dic P errin quaternions, and to examine the existence of zero divisors in the quater- nion algebra Q Z p . Understanding zero divisors in such split quaternion algebras ov er finite fields is a prob- lem of in trinsic algebraic in terest, linking num b er theoretic prop erties of sequences with the algebraic structure of quaternions. As a first step, we focus on the sp ecial mo dular setting where the twin prime co efficients a = p − 2 and b = p for a prime p ≥ 5. It is w orth emphasizing that the approac h adopted in the Fib onacci quaternion case studied b y Sa vin [19], as well as in the Leonardo quaternion setting considered in [21, 26], cannot b e directly transferred to the bi-p erio dic Pado v an framew ork. In those cases, the underlying recurrences lead to norm expressions with a relatively uniform structure, allowing zero divisors and in v ertible elemen ts to b e c haracterized without imp osing strong restrictions on the defining parameters. By con trast, the bi-p erio dic nature of the Pado v an and P er- rin sequences in tro duces alternating co efficients and non-consecutiv e index in teractions, whic h significan tly complicate the norm structure. This structural difference motiv ates the use of sp ecific co efficien t c hoices, suc h as twin prime pairs. On the other hand, in those pap ers the zero divisors were determined only for certain sp ecial v alues of the prime p , whereas the approac h adopted here applies to a general prime p , which we b elieve constitutes one of the main strengths of this pap er. In this context, w e explicitly de- termine which bi-p erio dic P ado v an and Perrin quaternions are zero divisors in Q Z p . W e then provide a detailed analysis of the zero divisor structure in the sp ecific cases p = 7, p = 13, and p = 181 for the bi-p erio dic P errin quaternions. In particular, w e show that for p = 13, all bi-p erio dic P errin quaternions are in v ertible. 2. Bi-periodic P adov an qua ternions over finite fields In this section, we in tro duce bi-p erio dic Pado v an quaternions and present some of their basic prop erties, including the recurrence relation and generating function. Then, b y 4 DIANA SA VIN AND ELIF T AN fixing a specific twin prime parameterization, we study their norm structure and deriv e criteria for the existence of zero divisors in Q Z p . Definition 1. The bi-p erio dic P adov an quaternion sequence { QP n } n ≥ 0 is defined as QP n = P n + P n +1 i + P n +2 j + P n +3 k where P n is the n -th bi-p erio dic P ado v an num b er. The first few terms can b e seen as follo ws: QP 0 = 1 + aj + k , QP 1 = ai + j + a 2 k , QP 2 = a + i + a 2 j + ( a + b ) k , QP 3 = 1 + a 2 i + ( a + b ) j + ( a 3 + 1) k , QP 4 = a 2 + ( a + b ) i + ( a 3 + 1) j + ( a 2 + ab + b 2 ) k , QP 5 = a + b + ( a 3 + 1) i + ( a 2 + ab + b 2 ) j + (( a + b ) + a ( a 3 + 1)) k . No w we derive the quaternion analogues of the iden tities satisfied by the bi-p erio dic P ado v an n umbers, as given in relations (1.2) and (1.3). The bi-p erio dic P ado v an quater- nions satisfy the follo wing recurrence relation: QP n = ( a + b ) QP n − 2 − ab QP n − 4 + QP n − 6 , n ≥ 6 . (2.1) F rom the recurrence relation (2.1), the generating function of the bi-p erio dic Pado v an quaternion sequence is giv en b y G ( x ) = ∞ X n =0 QP n x n = A ( x ) + B ( x ) i + C ( x ) j + D ( x ) k 1 − ( a + b ) x 2 + abx 4 − x 6 where A ( x ) = 1 − bx 2 + x 3 , B ( x ) = ax + x 2 − abx 3 + x 5 , C ( x ) = a + x − abx 2 + x 4 , D ( x ) = 1 + a 2 x + (1 − a 2 b ) x 3 + ax 5 . Next, we consider the quaternion algebra Q Z p = Q Z p ( − 1 , − 1) o ver the finite field Z p . W e fo cus on bi-p erio dic Pado v an quaternions with the t win prime co efficien t choice a = p − 2 and b = p , and analyze their norm structure in the mo dular setting. Using these norms, w e determine the zero divisors in Q Z p . F rom the definition of bi-p erio dic Pado v an sequence with co efficien ts a = p − 2 and b = p for o dd indices, we ha v e P 2 k +1 = pP 2 k − 1 + P 2 k − 2 ≡ P 2 k − 2 (mo d p ) . Replacing k b y ON BI-PERIODIC P ADO V AN AND PERRIN QUA TERNIONS 5 k + 1 yields P 2 k ≡ P 2 k +3 (mo d p ) (2.2) W e begin with a useful result that will help us determine the zero divisors. T o do this, first w e give the follo wing lemma. Lemma 1. F or the bi-p erio dic Padovan se quenc e with twin prime c o efficients a = p − 2 and b = p , we have P 2 k ≡ ( − 1) k ⌊ k/ 3 ⌋ X i =0 ( − 1) i  k − 2 i i  2 k − 3 i (mo d p ) . Pr o of. The pro of can b e done by mathematical induction o v er k . F or initial terms we ha v e P 0 ≡ 1 (mo d p ), P 2 ≡ p − 2 (mo d p ), and P 4 ≡ ( p − 2) 2 (mo d p ), which agree with the stated form ula. Assume that for some k ≥ 2, the statement holds for ev ery in teger j ≤ k . Now w e prov e that the formula holds for k + 1. Using the recurrence relation of bi-p erio dic Pado v an sequences for even indices, w e hav e P 2 k +2 ≡ − 2 P 2 k + P 2 k − 1 (mo d p ). F rom (2.2), w e hav e P 2 k − 1 ≡ P 2 k − 4 (mo d p ). Thus, we obtain P 2 k +2 ≡ − 2 P 2 k + P 2 k − 4 (mo d p ) . No w applying the induction h yp othesis, w e get P 2 k +2 ≡ ( − 1) k +1 ⌊ k/ 3 ⌋ X i =0 ( − 1) i  k − 2 i i  2 k +1 − 3 i + ( − 1) k ⌊ ( k − 2) / 3 ⌋ X i =0 ( − 1) i  k − 2 − 2 i i  2 k − 2 − 3 i (mo d p ) . By replacing i by i − 1 in the second sum and using P ascal’s identit y  k +1 − 2 i i  =  k − 2 i i  +  k − 2 i i − 1  , w e obtain P 2 k +2 ≡ ( − 1) k +1 ⌊ ( k +1) / 3 ⌋ X i =0 ( − 1) i  k + 1 − 2 i i  2 k +1 − 3 i (mo d p ) , whic h completes the pro of. □ Prop osition 1. F or the bi-p erio dic Padovan se quenc e with twin prime c o efficients a = p − 2 and b = p , we have P m ≡ ( ( − 1) k ( F k +3 − 1) (mo d p ) , if m = 2 k , ( − 1) k − 1 ( F k +2 − 1) (mo d p ) , if m = 2 k + 1 . 6 DIANA SA VIN AND ELIF T AN Pr o of. If m = 2 k , from Lemma 1 and from the recurrence relation of the Fib onacci sequence, w e hav e ⌊ k/ 3 ⌋ X i =0 ( − 1) i  k − 2 i i  2 k − 3 i = F k +3 − 1 . Th us, w e get P 2 k ≡ ( − 1) k ( F k +3 − 1) (mo d p ) . Similarly , for m = 2 k + 1, using (2.2), w e hav e P 2 k +1 ≡ P 2 k − 2 (mo d p ) , whic h immedi- ately yields the desired result. □ W e recall below some kno wn results concerning the entry p oin t and the Pisana perio d, whic h will b e used throughout the pap er. These definitions and results can b e found in [14, 19, 28, 29]. Let p b e a p ositive o dd prime. If F z is the smallest non-zero Fib onacci n umber with the prop ert y p | F z , then z = z ( p ) is defined as the entry p oint of p in the Fib onacci sequence. F or a p ositiv e in teger m , the Pisano p erio d π ( m ) is defined as the smallest p ositiv e in teger k such that F k ≡ 0 (mo d m ) and F k +1 ≡ 1 (mo d m ). Lemma 2. The fol lowing statements hold: (i) z ( p ) | π ( p ) ; (ii) π ( p ) = z ( p ) if z ( p ) ≡ 2 (mo d 4) ; (iii) π ( p ) = 2 z ( p ) if z ( p ) ≡ 0 (mo d 4) ; (iv) π ( p ) = 4 z ( p ) if z ( p ) is o dd; (v) p | F m if and only if z ( p ) | m , for al l m ∈ N . W e are no w ready to state one of our main results, which giv es a complete characteri- zation of when a bi-p erio dic Pado v an quaternion with twin prime coefficients b ecomes a zero divisor in Q Z p . Theorem 1. L et m b e an even inte ger and m 2 ≡ − 3 (mo d z ( p )) . Then, the bi-p erio dic Padovan quaternion QP m with twin prime c o efficients a = p − 2 and b = p is a zer o divisor in the quaternion algebr a Q Z p if and only if p ≡ 1 (mo d 4) and m 2 ≡ s − 2 (mo d π ( p )) , wher e s ∈ { z ( p ) − 1 , 2 z ( p ) − 1 , 3 z ( p ) − 1 , 4 z ( p ) − 1 } . Pr o of. If m is even, then m = 2 k , k ∈ N . A bi-p erio dic Pado v an quaternion QP m is a zero divisor in the quaternion algebra Q Z p if and ony if the norm N ( QP m ) ≡ 0 ( mo d p ) . Thus w e ha v e N ( QP m ) ≡ 0 ( mo d p ) ⇔ P 2 m + P 2 m +1 + P 2 m +2 + P 2 m +3 ≡ 0 ( mo d p ) ⇔ P 2 2 k + P 2 2 k +1 + P 2 2 k +2 + P 2 2 k +3 ≡ 0 ( mo d p ) . ON BI-PERIODIC P ADO V AN AND PERRIN QUA TERNIONS 7 Using Proposition 1, w e obtain N ( QP m ) ≡ 0 ( mo d p ) ⇔  ( − 1) k ( F k +3 − 1)  2 +  ( − 1) k − 1 ( F k +2 − 1)  2 +  ( − 1) k +1 ( F k +4 − 1)  2 +  ( − 1) k ( F k +3 − 1)  2 ≡ 0 ( mo d p ) ⇔ 2 ( F k +3 − 1) 2 + ( F k +2 − 1) 2 + ( F k +4 − 1) 2 ≡ 0 ( mo d p ) . Since m 2 ≡ − 3 ( mo d z ( p )) and from Lemma 2(v), it follows z ( p ) | k + 3 ⇔ p | F k +3 ⇔ F k +2 ≡ F k +4 ( mo d p ) (2.3) F rom here, we obtain that ( F k +2 − 1) 2 ≡ ( F k +4 − 1) 2 ( mo d p ). Also, since p | F k +3 , F 2 k +3 ≡ 0 ( mo d p ). So, we get N ( QP m ) ≡ 0 ( mo d p ) ⇔ ( F k +3 − 1) 2 + ( F k +2 − 1) 2 ≡ 0 ( mo d p ) ⇔ ( F k +2 − 1) 2 ≡ − 1 ( mo d p ) ⇔  − 1 p  = 1 ⇔ p ≡ 1 ( mo d 4) . Note that the Legendre sym b ol  − 1 p  = ( − 1) p − 1 2 = 1 ⇔ p ≡ 1 ( mo d 4) . Moreo v er, a solution of the congruence x 2 ≡ − 1 ( mo d p ) is F k +2 − 1 . Let α b e an in teger solution of this congruence suc h that F k +2 ≡ α + 1 ( mo d p ). Then there exist s ∈ { 0 , 1 , . . . , π ( p ) − 1 } suc h that k ≡ s − 2 (mo d π ( p )). F rom the hypothesis, w e ha v e m 2 ≡ − 3 (mo d z ( p )) ⇔ k ≡ − 3 (mo d z ( p )) . By Lemma 2, we ha v e z ( p ) | π ( p ) , in fact π ( p ) ∈ { z ( p ) , 2 z ( p ) , 4 z ( p ) } . Hence, s − 2 ≡ − 3 (mo d z ( p )) ⇔ s ≡ z ( p ) − 1 (mo d z ( p )) . It follo ws that s ∈ { z ( p ) − 1 , 2 z ( p ) − 1 , 3 z ( p ) − 1 , 4 z ( p ) − 1 } . □ W e no w state the corresp onding result for o dd m . Theorem 2. L et m b e an o dd inte ger and m − 1 2 ≡ − 3 (mo d z ( p )) . Then, the bi-p erio dic Padovan quaternion QP m with twin prime c o efficients a = p − 2 and b = p is a zer o divisor in the quaternion algebr a Q Z p if and only if p ≡ 1 (mo d 3) and m − 1 2 ≡ s − 2 (mo d π ( p )) , wher e s ∈ { z ( p ) − 1 , 2 z ( p ) − 1 , 3 z ( p ) − 1 , 4 z ( p ) − 1 } . Pr o of. If m is o dd, then m = 2 k + 1 , k ∈ N . Since m − 1 2 ≡ − 3 ( mo d z ( p )), w e hav e k ≡ − 3 ( mo d z ( p )). F rom Lemma 2 (v), it follo ws z ( p ) | k + 3 ⇔ p | F k +3 ⇔ F k +3 ≡ 0 ( mo d p ) (2.4) 8 DIANA SA VIN AND ELIF T AN A bi-perio dic P ado v an quaternion QP m is a zero divisor in the quaternion algebra Q Z p if and only if the norm N ( QP m ) ≡ 0 ( mo d p ). Th us w e ha v e N ( QP m ) ≡ 0 ( mo d p ) ⇔ P 2 m + P 2 m +1 + P 2 m +2 + P 2 m +3 ≡ 0 ( mo d p ) ⇔ P 2 2 k +1 + P 2 2 k +2 + P 2 2 k +3 + P 2 2 k +4 ≡ 0 ( mo d p ) ⇔ P 2 2 k − 2 + P 2 2 k +2 + P 2 2 k + P 2 2 k +4 ≡ 0 ( mo d p ) . Using Proposition 1, w e obtain N ( QP m ) ≡ 0 ( mo d p ) ⇔  ( − 1) k − 1 ( F k +2 − 1)  2 +  ( − 1) k ( F k +3 − 1)  2 +  ( − 1) k +1 ( F k +4 − 1)  2 +  ( − 1) k +2 ( F k +5 − 1)  2 ≡ 0 ( mo d p ) ⇔ ( F k +2 − 1) 2 + ( F k +3 − 1) 2 + ( F k +3 + F k +2 − 1) 2 + (2 F k +3 + F k +2 − 1) 2 ≡ 0 ( mo d p ) . F rom (2.4), w e hav e N ( QP m ) ≡ 0 (mo d p ) ⇔ 3 ( F k +2 − 1) 2 ≡ − 1 (mo d p ) ⇔ (3 ( F k +2 − 1)) 2 ≡ − 3 (mo d p ) ⇔  − 3 p  = 1 ⇔  − 1 p  3 p  = 1 ⇔  3 p  = ( − 1) p − 1 2 . Since p is a prime with p ≥ 5 , b y the quadratic recipro cit y la w, w e ha v e  3 p  p 3  = ( − 1) p − 1 2 ⇔  3 p  = ( − 1) p − 1 2  p 3  . Therefore, w e obtain N ( QP m ) ≡ 0 ( mo d p ) ⇔  p 3  = 1 ⇔ p ≡ 1 ( mo d 3) . Moreo v er, a solution of the congruence x 2 ≡ − 3 ( mo d p ) is 3 ( F k +2 − 1) . Let α be an in teger solution of the congruence x 2 ≡ − 3 ( mo d p ) suc h that 3 ( F k +2 − 1) ≡ α ( mo d p ) . Since p  = 3, the integer 3 is inv ertible modulo p ; let β denote its in v erse modulo p . It then follo ws that F k +2 ≡ β α +1 (mo d p ). Consequently , there exist s ∈ { 0 , 1 , . . . , π ( p ) − 1 } such that k ≡ s − 2 (mo d π ( p )). F rom the hypothesis, we hav e m − 1 2 ≡ − 3 (mo d z ( p )) ⇔ k ≡ ON BI-PERIODIC P ADO V AN AND PERRIN QUA TERNIONS 9 − 3 (mo d z ( p )) . By Lemma 2, we ha v e z ( p ) | π ( p ), and in fact π ( p ) ∈ { z ( p ) , 2 z ( p ) , 4 z ( p ) } . Hence, s − 2 ≡ − 3 (mo d z ( p )) ⇔ s ≡ z ( p ) − 1 (mo d z ( p )) . It follo ws that s ∈ { z ( p ) − 1 , 2 z ( p ) − 1 , 3 z ( p ) − 1 , 4 z ( p ) − 1 } . □ 3. Bi-periodic Perrin qua ternions over finite fields In this section, we in tro duce the bi-perio dic P errin sequence and presen t its relationship with the bi-p erio dic Pado v an sequence. Then, we define the bi-p erio dic P errin quater- nions and deriv e the corresp onding quaternion iden tities. Finally , b y fixing a sp ecific t win prime parameterization, w e study their norm structure and obtain criteria for the existence of zero divisors in Q Z p . Let { P n ( a, b ) } n ≥ 0 denote the bi-p erio dic Pado v an sequence { P n } n ≥ 0 defined in [1]. Using the same recurrence structure, w e now introduce the bi-p erio dic P errin sequence. Definition 2. The bi-p erio dic P errin sequence { R n ( a, b ) } n ≥ 0 , or simply { R n } n ≥ 0 , is de- fined b y the recurrence relation R n ( a, b ) = ( a R n − 2 ( a, b ) + R n − 3 ( a, b ) , if n is even , b R n − 2 ( a, b ) + R n − 3 ( a, b ) , if n is o dd , n ≥ 3 , with initial v alues R 0 ( a, b ) = 3 , R 1 ( a, b ) = 0 , and R 2 ( a, b ) = 2 . The first few terms of these sequences are given in T able 1. n P n R n 0 1 3 1 0 0 2 a 2 3 1 3 4 a 2 2 a 5 a + b 2 + 3 b 6 1 + a 3 3 + 2 a 2 7 a 2 + ab + b 2 2 a + 2 b + 3 b 2 8 a 4 + 2 a + b 2 a 3 + 3 a + 3 b + 2 9 1 + a 3 + a 2 b + ab 2 + b 3 3 + 2 a 2 + 2 ab + 2 b 2 + 3 b 3 10 a 5 + 3 a 2 + 2 ab + b 2 2 a 4 + 3 a 2 + 4 a + 3 ab + 2 b + 3 b 2 T able 1. The first terms of the bi-p erio dic P adov an and the bi-p erio dic P errin sequences. 10 DIANA SA VIN AND ELIF T AN Unlik e the classical case, the relation b etw een the bi-p erio dic P adov an and P errin se- quences depends on the parity of the index. Prop osition 2. F or n ≥ 3 , the bi-p erio dic Padovan se quenc e and bi-p erio dic Perrin se quenc e satisfy the fol lowing r elation: R n ( a, b ) = ( 3 P n − 3 ( a, b ) + 2 P n − 2 ( a, b ) , if n is even , 3 P n − 3 ( b, a ) + 2 P n − 2 ( b, a ) , if n is o dd , Pr o of. W e prov e the statement b y induction on n . The cases n = 3 and n = 4 are verified directly . No w assume that the statemen t holds for all indices up to n = 2 m , i.e., R 2 m ( a, b ) = 3 P 2 m − 3 ( a, b ) + 2 P 2 m − 2 ( a, b ) and R 2 m − 1 ( a, b ) = 3 P 2 m − 4 ( b, a ) + 2 P 2 m − 3 ( b, a ) . W e pro v e the statemen t for n = 2 m + 1 and n = 2 m + 2. Let n = 2 m + 1. Using the recurrence relation of the bi-p erio dic P errin sequence, we ha v e R 2 m +1 ( a, b ) = bR 2 m − 1 ( a, b ) + R 2 m − 2 ( a, b ) . Applying the induction h yp othesis, w e get R 2 m +1 ( a, b ) = b  3 P 2 m − 4 ( b, a ) + 2 P 2 m − 3 ( b, a )  + 3 P 2 m − 5 ( a, b ) + 2 P 2 m − 4 ( a, b ) . Since P 2 k +1 ( a, b ) = P 2 k +1 ( b, a ) for all k ≥ 0, and b y using the definition of the bi-p erio dic P ado v an sequence, w e obtain R 2 m +1 ( a, b ) = 3  bP 2 m − 4 ( b, a ) + P 2 m − 3 ( b, a )  + 2  bP 2 m − 3 ( a, b ) + P 2 m − 4 ( a, b )  = 3 P 2 m − 2 ( b, a ) + 2 P 2 m − 1 ( b, a ) . Let n = 2 m + 2. Using the recurrence relation of the bi-p erio dic P errin sequence, we ha v e R 2 m +2 ( a, b ) = aR 2 m ( a, b ) + R 2 m − 1 ( a, b ) . By the induction h yp othesis, w e get R 2 m +2 ( a, b ) = a  3 P 2 m − 3 ( a, b ) + 2 P 2 m − 2 ( a, b )  + 3 P 2 m − 4 ( b, a ) + 2 P 2 m − 3 ( b, a ) . Since P 2 k +1 ( a, b ) = P 2 k +1 ( b, a ) for all k ≥ 0, and b y using the definition of the bi-p erio dic P ado v an sequence, w e obtain R 2 m +2 ( a, b ) = 3  aP 2 m − 3 ( b, a ) + P 2 m − 4 ( b, a )  + 2  aP 2 m − 2 ( a, b ) + P 2 m − 3 ( a, b )  = 3 P 2 m − 1 ( a, b ) + 2 P 2 m ( a, b ) . This completes the pro of. □ T o derive a unified relation for the quaternion case, cov ering b oth the o dd and even cases, using the bi-perio dic P adov an and P errin sequences, w e no w define the bi-p erio dic P errin quaternions as follo ws. ON BI-PERIODIC P ADO V AN AND PERRIN QUA TERNIONS 11 Definition 3. The bi-p erio dic P errin quaternion sequence { QR n ( a, b ) } n ≥ 0 , or simply { QR n } n ≥ 0 , is defined as QR n ( a, b ) = ( R n ( a, b ) + R n +1 ( b, a ) i + R n +2 ( a, b ) j + R n +3 ( b, a ) k , if n is ev en , R n ( b, a ) + R n +1 ( a, b ) i + R n +2 ( b, a ) j + R n +3 ( a, b ) k , if n is o dd , where R n ( a, b ) is the n -th bi-p erio dic P errin n um b er. The first few terms of the bi-p erio dic P errin quaternion sequence are giv en b y QR 0 = 3 + 2 j + 3 k , QR 1 = 2 i + 3 j + 2 a k , QR 2 = 2 + 3 i + 2 a j + (3 a + 2) k , QR 3 = 3 + 2 a i + (3 a + 2) j + (2 a 2 + 3) k , QR 4 = 2 a + (3 a + 2) i + (2 a 2 + 3) j + (3 a 2 + 2 a + 2 b ) k , QR 5 = (3 a + 2) + (2 a 2 + 3) i + (3 a 2 + 2 a + 2 b ) j + (2 a 3 + 3 a + 3 b + 2) k . It is clear to see that bi-perio dic P errin quaternions also satisfy the same recurrence as bi-p erio dic P adov an quaternions given in (2.1). By considering this recurrence relation, the generating function of the bi-p erio dic P errin quaternion sequence is giv en by ∞ X n =0 QR n ( a, b ) x n = A ′ ( x ) + B ′ ( x ) i + C ′ ( x ) j + D ′ ( x ) k 1 − ( a + b ) x 2 + abx 4 − x 6 where A ′ ( x ) = 3 + (2 − 3 a − 3 b ) x 2 + 3 x 3 + b (3 a − 2) x 4 + (2 − 3 b ) x 5 , B ′ ( x ) = 2 x + 3 x 2 − 2 bx 3 + (2 − 3 b ) x 4 + 3 x 5 , C ′ ( x ) = 2 + 3 x − 2 bx 2 + (2 − 3 b ) x 3 + 3 x 4 , D ′ ( x ) = 3 + 2 ax + (2 − 3 b ) x 2 + (3 − 2 ab ) x 3 + 2 x 5 . Analogous to the relationship betw een the bi-p erio dic P ado v an and bi-p erio dic Perrin sequences, the corresponding connection also holds for their quaternion extensions. This follo ws directly from the definition of bi-p erio dic Perrin quaternions and Prop osition 2; hence, w e omit the pro of. Prop osition 3. F or al l n ≥ 3 , QR n ( a, b ) = 3 QP n − 3 ( a, b ) + 2 QP n − 2 ( a, b ) . In a similar manner, w e consider bi-p erio dic Perrin quaternions with t win prime co- efficien ts a = p − 2 and b = p and state another main result providing a complete 12 DIANA SA VIN AND ELIF T AN c haracterization of when a bi-p erio dic P errin quaternion with t win prime co efficien ts be- comes a zero divisor in Q Z p . Note that w e exclude the case p = 181 since in this case the discriminan t of the quadratic congruence v anishes modulo p . Theorem 3. L et m b e even and m 2 ≡ − 3 (mo d z ( p )) . Then, the bi-p erio dic Perrin quaternion QR m ( a, b ) with twin prime c o efficients a = p − 2 and b = p for primes p  = 181 , is a zer o divisor in the quaternion algebr a Q Z p if and only if m 2 ≡ s − 1 (mo d π ( p )) , s ∈ { z ( p ) − 2 , 2 z ( p ) − 2 , 3 z ( p ) − 2 , 4 z ( p ) − 2 } , and one of the fol lowing holds: (i) p ≡ 1 , 3 (mo d 8) and p is a quadr atic r esidue mo dulo 181 ; (ii) p ≡ 5 , 7 (mo d 8) and p is a quadr atic non-r esidue mo dulo 181 . Pr o of. If m is ev en, then m = 2 k , k ∈ N . A bi-p erio dic P errin quaternion QR m ( p − 2 , p ) is a zero divisor in the quaternion algebra Q Z p if and only if its norm satisfies N ( QR m ( p − 2 , p )) ≡ 0 ( mo d p ) . By the definition of the norm of a quaternion and Definition 3, we hav e N ( QR m ) ≡ 0 ( mo d p ) ⇔ R 2 m ( p − 2 , p ) + R 2 m +1 ( p, p − 2) + R 2 m +2 ( p − 2 , p ) + R 2 m +3 ( p, p − 2) ≡ 0 ( mo d p ) ⇔ R 2 2 k ( p − 2 , p ) + R 2 2 k +1 ( p, p − 2) + R 2 2 k +2 ( p − 2 , p ) + R 2 2 k +3 ( p, p − 2) ≡ 0 ( mo d p ) . Using Proposition 3, w e obtain N ( QR m ) ≡ 0 ( mo d p ) ⇔ (3 P 2 k − 3 + 2 P 2 k − 2 ) 2 + (3 P 2 k − 2 + 2 P 2 k − 1 ) 2 + (3 P 2 k − 1 + 2 P 2 k ) 2 + (3 P 2 k + 2 P 2 k +1 ) 2 ≡ 0 ( mo d p ) ⇔ 9  P 2 2 k − 3 + P 2 2 k − 2 + P 2 2 k − 1 + P 2 2 k  + 4  P 2 2 k − 2 + P 2 2 k − 1 + P 2 2 k + P 2 2 k +1  + 12 ( P 2 k − 3 P 2 k − 2 + P 2 k − 2 P 2 k − 1 + P 2 k − 1 P 2 k + P 2 k P 2 k +1 ) ≡ 0 ( mo d p ) ⇔ 13  P 2 2 k − 2 + P 2 2 k − 1 + P 2 2 k  + 9 P 2 2 k − 3 + 4 P 2 2 k +1 + 12 ( P 2 k − 3 P 2 k − 2 + P 2 k − 2 P 2 k − 1 + P 2 k − 1 P 2 k + P 2 k P 2 k +1 ) ≡ 0 ( mo d p ) . Using (2.2), we hav e N ( QR m ) ≡ 0 ( mo d p ) ⇔ 13  P 2 2 k − 2 + P 2 2 k − 4 + P 2 2 k  + 9 P 2 2 k − 6 + 4 P 2 2 k − 2 + 12 ( P 2 k − 6 P 2 k − 2 + P 2 k − 2 P 2 k − 4 + P 2 k − 4 P 2 k + P 2 k P 2 k − 2 ) ≡ 0 ( mo d p ) . ON BI-PERIODIC P ADO V AN AND PERRIN QUA TERNIONS 13 Using Proposition (1), w e obtain N ( QR m ) ≡ 0 (mo d p ) ⇔ 17  ( − 1) k − 1 ( F k +2 − 1)  2 + 13  ( − 1) k − 2 ( F k +1 − 1)  2 + 13  ( − 1) k ( F k +3 − 1)  2 + 9  ( − 1) k − 3 ( F k − 1)  2 + 12  ( − 1) k ( F k − 1)   ( − 1) k − 1 ( F k +2 − 1)  + 12  ( − 1) k − 1 ( F k +2 − 1)   ( − 1) k − 2 ( F k +1 − 1)  + 12  ( − 1) k − 2 ( F k +1 − 1)   ( − 1) k ( F k +3 − 1)  + 12  ( − 1) k ( F k +3 − 1)   ( − 1) k − 1 ( F k +2 − 1)  ≡ 0 (mo d p ) . Since m 2 ≡ − 3 (mo d z ( p )) , w e hav e z ( p ) | ( k + 3) . By Lemma 2(v), this is equiv alent to p | F k +3 . Therefore, we obtain N ( QR m ) ≡ 0 ( mo d p ) ⇔ 17 ( F k +2 − 1) 2 + 13 ( F k +1 − 1) 2 + 13 + 9 ( F k − 1) 2 − 12 ( F k − 1) ( F k +2 − 1) − 12 ( F k +2 − 1) ( F k +1 − 1) − 12 ( F k +1 − 1) + 12 ( F k +2 − 1) ≡ 0 ( mo d p ) . Since p | F k +3 , w e hav e F k +3 ≡ 0 (mo d p ) . By the recurrence relation of the Fib onacci sequence, it follows that F k +3 ≡ 0 (mo d p ) ⇔ F k +2 ≡ − F k +1 ( mo d p ) , and also F k +3 ≡ 0 ( mo d p ) ⇔ F k ≡ − 2 F k +1 ( mo d p ). Th us, w e obtain N ( QR m ) ≡ 0 ( mo d p ) ⇔ 17 ( F k +1 + 1) 2 + 13 ( F k +1 − 1) 2 + 13 + 9 (2 F k +1 + 1) 2 − 12 ( − 2 F k +1 − 1) ( − F k +1 − 1) − 12 ( − F k +1 − 1) ( F k +1 − 1) − 12 ( F k +1 − 1) + 12 ( − F k +1 − 1) ≡ 0 ( mo d p ) ⇔ 54 F 2 k +1 − 16 F k +1 + 28 ≡ 0 (mo d p ) . Since p is a prime with p  = 2, w e ha v e N ( QR m ) ≡ 0 (mo d p ) ⇔ 27 F 2 k +1 − 8 F k +1 + 14 ≡ 0 (mo d p ) . The congruence 27 x 2 − 8 x + 14 ≡ 0 (mo d p ) (3.1) has an in teger solution mo dulo p if and only if its discriminant ∆ = − 8 · 181 is a quadratic residue mo dulo p , i.e.,  ∆ p  =  − 8 · 181 p  = 1 , see [17]. Using the m ultiplicativity of the 14 DIANA SA VIN AND ELIF T AN Legendre sym bol, we ha ve  − 8 · 181 p  =  − 1 p   2 p   181 p  . Recalling that  − 1 p  = ( − 1) p − 1 2 and  2 p  = ( − 1) p 2 − 1 8 , w e obtain  181 p  = ( − 1) p 2 − 1 8 ( − 1) p − 1 2 = ( − 1) ( p − 1)( p +5) 8 . Finally , applying the quadratic recipro cit y law, the congruence has a solution if and only if  p 181  = ( − 1) ( p − 1)( p +5) 8 . This yields the follo wing explicit conditions:  ∆ p  = 1 ⇔      p ≡ 1 , 3 (mo d 8) and p is a quadratic residue mo dulo 181 , or p ≡ 5 , 7 (mo d 8) and p is a quadratic non-residue mo dulo 181 . Moreo v er, a solution of the congruence (3.1) is F k +1 , so we hav e 27 F 2 k +1 − 8 F k +1 + 14 ≡ 0 (mo d p ) ⇔ (2 · 27 F k +1 − 8) 2 ≡ ∆ (mo d p ) ⇔ (2 · 27 F k +1 − 8) 2 ≡ − 8 · 181 (mo d p ) ⇔ (27 F k +1 − 4) 2 ≡ − 2 · 181 (mo d p ) . Let α b e an integer solution of the congruence x 2 ≡ − 2 · 181 ( mo d p ) such that 27 F k +1 ≡ α + 4 ( mo d p ) . Since p  = 3, the in teger 27 is in v ertible mo dulo p ; let β denote its in v erse. It then follows that F k +1 ≡ β ( α + 4) (mo d p ). Therefore, there exists an integer s ∈ { 0 , 1 , . . . , π ( p ) − 1 } suc h that k ≡ s − 1 (mo d π ( p )). On the other hand, from the h yp othesis, we ha v e m 2 ≡ − 3 (mo d z ( p )) ⇔ k ≡ − 3 (mo d z ( p )) . By Lemma 2, it results z ( p ) | π ( p ), and in fact π ( p ) ∈ { z ( p ) , 2 z ( p ) , 4 z ( p ) } . Therefore, we ha ve s − 1 ≡ − 3 (mo d z ( p )) ⇔ s ≡ z ( p ) − 2 (mo d z ( p )) . It follo ws that s ∈ { z ( p ) − 2 , 2 z ( p ) − 2 , 3 z ( p ) − 2 , 4 z ( p ) − 2 } . □ W e no w consider the case p = 181 in Theorem 3. Corollary 1. L et m b e an even p ositive inte ger such that m 2 ≡ − 3 (mo d z (181)) . Then the bi-p erio dic Perrin quaternion QR m ( a, b ) with twin prime c o efficients a = 179 and b = 181 is a zer o divisor in the quaternion algebr a Q Z 181 if and only if m 2 ≡ 47 ( mo d 90) . ON BI-PERIODIC P ADO V AN AND PERRIN QUA TERNIONS 15 Pr o of. If m is ev en, then m = 2 k, k ∈ N . Analogously as in the pro of of Theorem 3, w e obtain that the bi-perio dic P errin quaternion QR m (179 , 181) is a zero divisor in the quaternion algebra Q Z 181 if and only if (27 F k +1 − 4) 2 ≡ 0 ( mo d 181) ⇔ 27 F k +1 ≡ 4 ( mo d 181) . Since gcd(27 , 181) = 1, it follo ws that the last congruence has a unique solution mo dulo 181 and this is 4 · 27 φ (181) ( mo d 181) = 4 · 27 180 ( mo d 181) ≡ 4 · 114 ( mo d 181) , where φ is the Euler’s function. So, w e ha v e 27 F k +1 ≡ 4 ( mo d 181) ⇔ F k +1 ≡ 4 · 114 ( mo d 181) ⇔ F k +1 ≡ 94 ( mo d 181) . F rom [14], the Pisano p erio d π (181) = 90 and F k +1 ≡ 94 ( mo d 181) ⇔ k + 1 ≡ 48 ( mo d 90) ⇔ k ≡ 47 ( mo d 90) ⇔ m 2 ≡ 47 ( mo d 90) . □ W e no w state the corresp onding result for o dd m . Theorem 4. L et m b e an o dd inte ger and m − 1 2 ≡ − 3 (mo d z ( p )) . Then, the bi-p erio dic Perrin quaternion QR m ( a, b ) with twin prime c o efficients a = p − 2 and b = p for primes p  = 7 , 13 , is a zer o divisor in the quaternion algebr a Q Z p if and only if the Jac obi symb ol satisfies  p 13 · 239  = 1 and m − 1 2 ≡ s − 1 (mo d π ( p )) , wher e s ∈ { z ( p ) − 2 , 2 z ( p ) − 2 , 3 z ( p ) − 2 , 4 z ( p ) − 2 } . Pr o of. If m is odd, then m = 2 k + 1 , k ∈ N . A bi-perio dic P errin quaternion QR m ( p − 2 , p ) is a zero divisor in the quaternion algebra Q Z p if and ony if the norm N ( QR m ( p − 2 , p )) ≡ 0 ( mo d p ) . By the definition of the norm of a quaternion and Definition 3, we hav e N ( QR m ) ≡ ( mo d p ) ⇔ R 2 m ( p, p − 2) + R 2 m +1 ( p − 2 , p ) + R 2 m +2 ( p, p − 2) + R 2 m +3 ( p − 2 , p ) ≡ 0 ( mo d p ) ⇔ R 2 2 k +1 ( p, p − 2) + R 2 2 k +2 ( p − 2 , p ) + R 2 2 k +3 ( p, p − 2) + R 2 2 k +4 ( p − 2 , p ) ≡ 0 ( mo d p ) . 16 DIANA SA VIN AND ELIF T AN Using Proposition 3, w e obtain N ( QR m ) ≡ 0 ( mo d p ) ⇔ (3 P 2 k − 2 + 2 P 2 k − 1 ) 2 + (3 P 2 k − 1 + 2 P 2 k ) 2 + (3 P 2 k + 2 P 2 k +1 ) 2 + (3 P 2 k +1 + 2 P 2 k +2 ) 2 ≡ 0 ( mo d p ) ⇔ 9  P 2 2 k − 2 + P 2 2 k − 1 + P 2 2 k + P 2 2 k +1  + 4  P 2 2 k − 1 + P 2 2 k + P 2 2 k +1 + P 2 2 k +2  + 12 ( P 2 k − 2 P 2 k − 1 + P 2 k − 1 P 2 k + P 2 k P 2 k +1 + P 2 k +1 P 2 k +2 ) ≡ 0 ( mo d p ) ⇔ 13  P 2 2 k − 1 + P 2 2 k + P 2 2 k +1  + 9 P 2 2 k − 2 + 4 P 2 2 k +2 + 12 ( P 2 k − 2 P 2 k − 1 + P 2 k − 1 P 2 k + P 2 k P 2 k +1 + P 2 k +1 P 2 k +2 ) ≡ 0 ( mo d p ) . Using (2.2), we hav e N ( QR m ) ≡ 0 ( mo d p ) ⇔ 13  P 2 2 k − 4 + P 2 2 k + P 2 2 k − 2  + 9 P 2 2 k − 2 + 4 P 2 2 k +2 + +12 ( P 2 k − 2 P 2 k − 4 + P 2 k − 4 P 2 k + P 2 k P 2 k − 2 + P 2 k − 2 P 2 k +2 ) ≡ 0 ( mo d p ) . Using Proposition (1), w e obtain N ( QR m ) ≡ 0 ( mo d p ) ⇔ 13 ( F k +1 − 1) 2 + 22 ( F k +2 − 1) 2 + 13 ( F k +3 − 1) 2 + 4 ( F k +4 − 1) 2 − 12 ( F k +2 − 1) ( F k +1 − 1) + 12 ( F k +1 − 1) ( F k +3 − 1) − 12 ( F k +3 − 1) ( F k +2 − 1) + 12 ( F k +2 − 1) ( F k +4 − 1) ≡ 0 ( mo d p ) . Since m − 1 2 ≡ − 3 (mo d z ( p )), w e hav e z ( p ) | ( k + 3). By Lemma 2(v), this is equiv alent to p | F k +3 . Therefore, we obtain N ( QR m ) ≡ 0 ( mo d p ) ⇔ 13 ( F k +1 − 1) 2 + 22 ( F k +2 − 1) 2 + 13 + 4 ( F k +4 − 1) 2 − 12 ( F k +2 − 1) ( F k +1 − 1) − 12 ( F k +1 − 1) + 12 ( F k +2 − 1) + 12 ( F k +2 − 1) ( F k +4 − 1) ≡ 0 ( mo d p ) . Since p | F k +3 , we hav e F k +3 ≡ 0 ( mo d p ). F rom the recurrence relation of the Fib onacci sequence, it follows that F k +3 ≡ 0 ( mo d p ) ⇔ F k +2 ≡ − F k +1 ( mo d p ). Then F k +4 ≡ − F k +1 ( mo d p ). Th us, w e obtain N ( QR m ) ≡ 0 ( mo d p ) ⇔ 13 ( F k +1 − 1) 2 + 22 ( F k +1 + 1) 2 + 13 + 4 ( F k +1 + 1) 2 − 12 ( − F k +1 − 1) ( F k +1 − 1) − 12 ( F k +1 − 1) + 12 ( − F k +1 − 1) + 12 ( − F k +1 − 1) ( − F k +1 − 1) ≡ 0 ( mo d p ) ⇔ 63 F 2 k +1 + 26 F k +1 + 52 ≡ 0 ( mo d p ) . The congruence 63 x 2 + 26 x + 52 ≡ 0 ( mo d p ) (3.2) ON BI-PERIODIC P ADO V AN AND PERRIN QUA TERNIONS 17 has an integer solution mo dulo p if and only if its discriminant ∆ = − 4 · 13 · 239 is a quadratic residue mo dulo p , i.e.,  ∆ p  =  − 4 · 13 · 239 p  = 1 , see [17]. Note that p  = 239, since in that case p = 239 and p − 2 = 237 w ould not form a t win prime pair. Using the m ultiplicativit y of the Legendre sym b ol, w e ha v e  − 4 · 13 · 239 p  = ( − 1) p − 1 2  13 p   239 p  . Using the quadratic recipro city la w, we ha ve  13 p  =  p 13  and  239 p  = ( − 1) p − 1 2  p 239  . By com bining these results, w e obtain  ∆ p  =  p 13   p 239  . So,  ∆ p  = 1 ⇔  p 13   p 239  = 1 . Using the fact that the Jacobi sym b ol  p 13 · 239  is a pro duct of t w o Legendre symbols, namely  p 13 · 239  =  p 13   p 239  , w e obtain  ∆ p  = 1 ⇔  p 13 · 239  = 1 . Moreo v er, a solution of the congruence (3.2) is F k +1 , so we hav e 63 F 2 k +1 + 26 F k +1 + 52 ≡ 0 ( mo d p ) ⇔ (2 · 63 · F k +1 + 26) 2 ≡ − 4 · 13 · 239 ( mo d p ) ⇔ (63 F k +1 + 13) 2 ≡ − 13 · 239 ( mo d p ) ⇔ (63 F k +1 + 13) 2 ≡ − 3107 ( mo d p ) . Let α be an in teger solution of the congruence x 2 ≡ − 3107 ( mo d p ) suc h that 63 F k +1 ≡ α − 13 ( mo d p ) . Since p  = 3 and p  = 7, the integer 63 is in v ertible modulo p . Let β denote its in v erse mo dulo p . Then it follows that F k +1 ≡ β ( α − 13) (mo d p ) . Consequently , there exists an in teger s ∈ { 0 , 1 , . . . , π ( p ) − 1 } suc h that k ≡ s − 1 (mo d π ( p )) . F rom the h yp othesis, w e hav e m − 1 2 ≡ − 3 (mo d z ( p )) ⇔ k ≡ − 3 (mo d z ( p )) . By Lemma 2, it results z ( p ) | π ( p ), and in fact π ( p ) ∈ { z ( p ) , 2 z ( p ) , 4 z ( p ) } . Hence, w e ha v e s − 1 ≡ − 3 (mo d z ( p )) ⇔ s ≡ z ( p ) − 2 (mo d z ( p )) . It follo ws that s ∈ { z ( p ) − 2 , 2 z ( p ) − 2 , 3 z ( p ) − 2 , 4 z ( p ) − 2 } . □ W e no w consider the case p = 7 in Theorem 4. Corollary 2. L et m b e an o dd inte ger such that m − 1 2 ≡ − 3 (mo d z (7)) . Then the bi- p erio dic Perrin quaternion QR m (5 , 7) is a zer o divisor in the quaternion algebr a Q Z 7 if and only if m − 1 2 ≡ 4 , 10 ( mo d 16) . 18 DIANA SA VIN AND ELIF T AN Pr o of. If m is o dd, then m = 2 k + 1 , k ∈ N . Analogously as in the proof of Theorem 4, w e obtain that a bi-p erio dic Perrin quaternion QR m (5 , 7) with is a zero divisor in the quaternion algebra Q Z 7 if and only if 63 F 2 k +1 + 26 F k +1 + 52 ≡ 0 ( mo d 7) ⇔ 5 F k +1 ≡ 4 ( mo d 7) . Since gcd(5 , 7) = 1, the last congruence is equiv alent to F k +1 ≡ 3 · 4 ( mo d 7) ⇔ F k +1 ≡ 5 ( mo d 7) . F rom [14], w e hav e k + 1 ≡ 5 , 11 ( mo d 16) ⇔ k ≡ 4 , 10 ( mo d 16) ⇔ m − 1 2 ≡ 4 , 10 ( mo d 16) . □ W e now examine the application of Theorem 4 to the case p = 13. Somewhat surpris- ingly , we show that in this setting all bi-p erio dic P errin quaternions are inv ertible in the quaternion algebra Q Z 13 . Corollary 3. L et m b e an o dd inte ger such that m − 1 2 ≡ − 3 (mo d z (13)) . Then al l bi- p erio dic Perrin quaternions QR m (11 , 13) ar e invertible in the quaternion algebr a Q Z 13 . Pr o of. If m is o dd, then m = 2 k + 1 , k ∈ N . It is kno wn that a quaternion is inv ertible in the quaternion algebra Q Z 13 if and only if it is not a zero divisor in the quaternion algebra Q Z 13 . Supp ose, b y con tradiction, that there exists a bi-p erio dic Perrin quaternion QR m (11 , 13) which is a zero divisor in the quaternion algebra Q Z 13 . Then, as in the pro of of Theorem 4, F k +1 m ust satisfy the congruence 3.2, i.e., 63 F 2 k +1 + 26 F k +1 + 52 ≡ 0 (mo d p ) . Reducing mo dulo 13, this simplifies to 11 F 2 k +1 ≡ 0 (mo d 13). Multiplying b oth sides by 6, the inv erse of 11 mo dulo 13, w e obtain F 2 k +1 ≡ 0 (mo d 13) whic h implies F k +1 ≡ 0 (mo d 13) . Th us, this is the only p ossible solution. F rom the hy p othesis, z (13) | ( k + 3), so 13 | F k +3 . By the Fib onacci recurrence, this implies 13 | F k +2 and then by induction, all Fib onacci n umbers w ould b e divisible by 13, whic h is impossible. Hence, our assumption is false and every QR m (11 , 13) is in v ertible in Q Z 13 . □ 4. Conclusion In this pap er, w e introduced the bi-perio dic P errin sequence and established its relation- ship with the bi-p erio dic Pado v an sequence. Extending these sequences to the quaternion setting, we analyzed their norms in the split algebra Q Z p . A key strength of our approach is that it applies to a general prime p , rather than b eing limited to sp ecific v alues. Within this setting, w e explicitly c haracterized which bi-p erio dic P ado v an and P errin quater- nions are zero divisors in Q Z p . In particular, for p = 13, we show ed that all bi-p erio dic ON BI-PERIODIC P ADO V AN AND PERRIN QUA TERNIONS 19 P errin quaternions of o dd index m , satisfying m − 1 2 ≡ − 3 (mo d z (13)) , are inv ertible in quaternion algebra Q Z p . As a direction for future research, it would b e of in terest to inv estigate quaternion algebras asso ciated with higher-dimensional k -p erio dic recurrence sequences for k > 2, where the bi-p erio dic case corresp onds to k = 2. A cknow ledgements The second author gratefully ackno wledges the T ransilv ania Univ ersit y of Bra¸ so v for supp orting her visit in Jan uary 2026. References [1] O. Dısk a ya and H. Menk en, On the bi-p erio dic P adov an sequences, Mathematic a Mor avic a , 27 (2) (2023), 115–126. [2] J. M. Grau, C. Miguel, and A. M. Oller-Marcen, On the structure of quaternion rings o v er Z /n Z , A dv. Appl. Cliffor d Algebr as , 25 (4) (2015), 875–887. [3] S. G ¨ ung¨ or, E. T an, and A. Belkhir, A c ombinatorial p ersp e ctive on gener alize d Padovan numb ers , submitted. [4] A. F. Horadam, Complex Fib onacci num b ers and Fib onacci quaternions, Amer. Math. Monthly , 70 (1963), 289–291. [5] S. Halici, On Fib onacci quaternions, A dv. Appl. Cliffor d Algebr as , 22 (2012), 321–327. [6] S. Halici and A. Karatas, On a generalization for quaternion sequences, Chaos Solitons F r actals , 98 (2017), 178–182. [7] Z. ˙ I¸ sbilir and N. G ¨ urses, Pado v an and P errin generalized quaternions, Math. Metho ds Appl. Sci. , 45 (18) (2022), 12060–12070. [8] T. Koshy , Fib onac ci and Luc as Numb ers with Applic ations , John Wiley & Sons, Hob oken, NJ, 2001. [9] T. Koshy , Fib onac ci and Luc as Numb ers with Applic ations, V ol. 2 , John Wiley & Sons, Hoboken, NJ, 2018. [10] G. Y. Lee, and P . Kisoeb, Pado v an and Lucas-Pado v an Quaternions, J. Kor e an So c. Math. Educ. Ser. B: Pur e Appl. Math. , 31 (4) (2024), 427-437. [11] C. J. Miguel and R. Sero dio, On the structure of quaternion rings ov er Z p , Int. J. Algebr a , 5 (27) (2011), 1313–1325. [12] The OEIS F oundation Inc., The On-Line Encyclop e dia of Inte ger Se quenc es , published electronically at https://oeis.org . [13] R. Pado v an, Mo dern Primitive: Dom Hans V an de L aan , Architectura & Natura Press, 1994. [14] M. Renault, The Fib onacci Sequence Under V arious Mo duli, Master’s Thesis, W ake F orest Univ er- sit y , 1996. [15] D.Savin, Division quaternion algebras ov er some cyclotomic fields, Communic ations in Algebr a , 53 (6) (2025), 2232-2246. [16] D. Savin, Some prop erties of Fibonacci num bers, Fib onacci o ctonions, and generalized Fib onacci- Lucas o ctonions, A dvanc es in Differ enc e Equations 2015 (298) (2015). [17] D. Sa vin and M. Stefanescu, L e ctur es on A rithmetic and Numb er The ory , Matrix Rom Publishing House, Buc harest, 2008. 20 DIANA SA VIN AND ELIF T AN [18] D. Savin, C. Flaut, and C. Ciobanu, Some prop erties of the sym b ol algebras, Carp athian J. Math. , 25 (2) (2009), 239–245. [19] D. Sa vin, Ab out sp ecial elements in quaternion algebras ov er finite fields, A dv. Appl. Cliffor d Alge- br as , 27 (2) (2017), 1801–1813. [20] D. Sa vin, About some split central simple algebras, A nalele Stiintific e ale Universitatii Ovidius Constanta , Ser. Mat. XXI I, f.1 (2014), p.263-272. [21] D. Savin and E. T an, On companion sequences asso ciated with Leonardo quaternions: Applications o ver finite fields, J. Inform. Optim. Sci. , 46 (5) (2025), 1569–1585. [22] E. T an, S. Yilmaz, and M. Sahin, On a new generalization of Fib onacci quaternions, Chaos Solitons F r actals , 82 (2016), 1–4. [23] E. T an, S. Yilmaz, and M. Sahin, A note on bi-p erio dic Fib onacci and Lucas quaternions, Chaos Solitons F r actals , 85 (2016), 138–142. [24] E. T an and H. H. Leung, Some results on Horadam quaternions, Chaos Solitons F r actals , 138 (2020), 109845. [25] E. T an and N. R. Ait-Amrane, On a new generalization of Fibonacci hybrid n um b ers, Indian J. Pur e Appl. Math. , 54 (2) (2023), 428–438. [26] E. T an, D. Savin, and S. Yilmaz, A new class of Leonardo h ybrid n um b ers and some remarks on Leonardo quaternions o ver finite fields, Mathematics , 11 (2023), 4701. [27] D. T asci, Pado v an and Pell–P ado v an quaternions, Journal of Scienc e and Arts , 18 (2018), 125–132. [28] D.D. W all, Fib onacci series mo dulo m. The Americ an Mathematic al Monthly , 67 (6) (1960), 525-532. [29] John Vinson, The Relation of the p erio d mo dulo m to the rank of apparition of m in the Fib onacci sequence, The Fib onac ci Quarterly , 1 (2) (1963), 37-46. Dep ar tment of Ma thema tics and Computer Science, Transil v ania University of Brasov, 500091, Romania Email addr ess : diana.savin@unitbv.ro; dianet72@yahoo.com Dep ar tment of Ma thema tics, F acul ty of Science, Ankara University 06100 T andogan Ankara, Turkey Email addr ess : etan@ankara.edu.tr