On the Order of Appearance of the Difference of Two Lucas Numbers

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Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 19 (2018), No. 1, pp. 641–648 DOI: 10.18514/MMN.2018.1750

ON THE ORDER OF APPEARANCE OF THE DIFFERENCE OF TWO LUCAS NUMBERS

PAVEL TROJOVSKY´

Received 22 September, 2015

Abstract. Let Fn be the nth Fibonacci number and let Ln be the nth Lucas number. The order of appearance ´.n/ of a natural number n is deﬁned as the smallest natural number k such that n divides Fk. For instance, ´.Ln/ 2n, for all n > 2. In this paper, among other things, we prove that D 2 2 5Fp m n ´.Lm Ln/ , D p 4 for all distinct positive integers m n .mod 4/, with gcd.m;n/ p > 2 prime. Á D 2010 Mathematics Subject Classiﬁcation: 11B39; 11A07 Keywords: Lucas number, Fibonacci number, order of appearance

1. INTRODUCTION

Let .Fn/n 0 be the Fibonacci sequence given by Fn 2 Fn 1 Fn, for n 0, C D C C where F0 0 and F1 1. These numbers are well-known for possessing amaz- ing propertiesD (consultD [4] together with its very extensive annotated bibliography for additional references and history). We cannot go very far in the lore of Fibon- acci numbers without encountering its companion Lucas sequence .Ln/n 0 which follows the same recursive pattern as the Fibonacci numbers, but with initial values L0 2 and L1 1. TheD study ofD the divisibility properties of Fibonacci numbers has always been a popular area of research. Let n be a positive integer number, the order (or rank) of appearance of n in the Fibonacci sequence, denoted by ´.n/, is deﬁned as the smallest positive integer k, such that n Fk (some authors also call it order of apparition, or Fibonacci entry point). There arej several results about ´.n/ in the literature. For instance, in 1975, J. Salle´ [13] proved that ´.n/ 2n, for all positive integers n. This is the sharpest upper bound for ´.n/, since forÄ example, ´.6/ 12 and ´.30/ 60 (see [10] for related results). In the case of a prime number p, oneD has the better upperD bound ´.p/ p 1, which is a consequence of the known congruence Fp . p / 0 Ä C 5 Á c 2018 Miskolc University Press

642 PAVEL TROJOVSKY´

.mod p/, for p 2, where . a / denotes the Legendre symbol of a with respect to a ¤ q prime q > 2. We remark that there is no a closed formula for ´.n/, but by using computational methods and several calculations, one can see patterns of ´.n/ for some positive integers n. For example, with this computational approach, Marques [7–9] found explicit formulas for the order of appearance of integers related to Fibonacci and k Lucas number, such as Cmk=Ck;Cm 1;CnCn 1Cn 2 and Cn , where .Cn/n ˙ C C D .Fn/n or .Ln/n. In this paper, we study the order of appearance of Lm Ln. Again, we used Mathematica to search for patterns for ´.Lm Ln/. We were surprised to find out that in some cases, these values are related to the sequences 1;2;5;13;89;233;1597;4181;28657;514229;::: and 3;4;11;29;199;521;3571;9349;64079;1149851;:::: A search in the On-Line Encyclopedia of Integer Sequences [14] is enough to conclude that, surprisingly, these sequences are the first few Fibonacci and Lucas numbers with prime indexes, respectively. In fact, this interpretation is confirmed by exhaustive calculations. More precisely, our main results are the following. Theorem 1. Let m and n > 1 be positive distinct integers, such that m n .mod 4/. We have Á (i) If gcd.m;n/ 1, then D 5.m2 n2/ ´.Lm Ln/ : D 4 (ii) If gcd.m;n/ p is prime, then D 2 2 5Fp m n ´.Lm Ln/ : D p 4

2. AUXILIARY RESULTS Before proceeding further, we recall some facts on Fibonacci and Lucas numbers for the convenience of the reader. Lemma 1. We have

(a) Fn Fm if and only if n m. j j (b) Ln Fm if and only if n m and m=n is even. j j (c) 5Fn F5n, for all integer n. j (d) If d gcd.m;n/, then gcd.Fm;Fn/ Fd , D D Ld ; if 2.m/ 2.n/ gcd.Lm;Ln/ D I D 1 or 2; otherwise ON THE ORDER OF APPEARANCE OF THE DIFFERENCE OF TWO LUCAS NUMBERS 643 Ld ; if m=d is even and n=d is odd and gcd.Fm;Ln/ I D 1 or 2; otherwise.

(e) Fp . 5 / 0 .mod p/, for all primes p. In particular, gcd.Fp;p/ 1, if p Á D p 5. ¤ (f) Lp 1 .mod p/, for all primes p. In particular, gcd.Lp;p/ 1. Á D a Here, as usual, . q / denotes the Legendre symbol of a with respect to a prime q > 2. Proof of these assertions can be found in [4]. We refer the reader to [1,3,6,11] for more details and additional bibliography. The second lemma is a consequence of the previous one Lemma 2 (Cf. Lemma 2.2 of [7]). We have

(a) If Fn m, then n ´.m/. j j (b) If Ln m, then 2n ´.m/. j j (c) If n Fm, then ´.n/ m. j j Lemma 3. For all primes p 5, we have that gcd.´.p/;p/ 1. ¤ D Proof. By combining Lemma1 (e) together with Lemma2 (c), we conclude that ´.p/ p . 5 /. Thus, when p 5, one has that . 5 / 1 and so ´.p/ divides j p ¤ p D ˙ p 1 or p 1. This yields that ´.p/ p 1 or ´.p/ p 1 and in any case gcd.´.p/;p/C 1. D C Ä D The next lemma will be very useful in the proof of our theorems. First, we recall the identities b FaLb Fa b . 1/ Fa b; D C C b 5FaFb La b . 1/ La b; (2.1) D C b LaLb La b . 1/ La b D C C for all integers a and b. These identities can be easily deduced from Binet’s formulas: n n ˛ ˇ n n Fn and Ln ˛ ˇ , for n 0, D ˛ ˇ D C where ˛ .1 p5/=2 and ˇ .1 p5/=2. D C D Lemma 4. Let m and n be positive integers, such that m n .mod 4/. Then Á (a) F.m n/=2L.m n/=2 Fm Fn: C D (b) F.m n/=2L.m n/=2 Fm Fn: C D C (c) L.m n/=2L.m n/=2 Lm Ln: C D C (d) 5F.m n/=2F.m n/=2 Lm Ln: C D 644 PAVEL TROJOVSKY´

Proof. It sufﬁces to use suitable choices of a and b in the identities (2.1). For example, if a .m n/=2 and b .m n/=2, then a b m and a b n. By (2.1), we haveD D C C D D m n C n 1 F.m n/=2L.m n/=2 Fm . 1/ 2 C C Fn; C D C n 1 where we used that F n . 1/ C Fn. The proof of (a) is complete, since .m n/=2 n 1 is odd. In fact,D this follows from C C C m n m n m n .mod 4/ C n .mod 2/ C n 1 2n 1 .mod 2/: Á ) 2 Á ) 2 C C Á C The other items can be proved in the same way.

The p-adic order (or valuation) of r, p.r/, is the exponent of the highest power of a prime p which divides r. Throughout the paper, we shall use the known facts that p.ab/ p.a/ p.b/ and that a b if and only if p.a/ p.b/, for all primes p. We remarkD thatC the p-adic order ofj Fibonacci and LucasÄ numbers was completely characterized, see [2,5, 12]. For instance, from the main results of Lengyel [5], we extract the following two results. Lemma 5. For n 1, we have 8 ˆ 0; if n 1;2 .mod 3/ <ˆ 1; if n Á 3 .mod 6/ I 2.Fn/ Á I D 3; if n 6 .mod 12/ ˆ Á I : 2.n/ 2; if n 0 .mod 12/; C Á 5.Fn/ 5.n/, and if p is prime 2 or 5, then D ¤ p.n/ p.F´.p//; if n 0 .mod ´.p// p.Fn/ C Á I D 0; otherwise. Lemma 6. Let k.p/ be the period modulo p of the Fibonacci sequence. For all primes p 5, we have ¤ 8 0; if n 1;2 .mod 3/ < Á I 2.Ln/ 2; if n 3 .mod 6/ D : 1; if n Á 0 .mod 6/I Á ´.p/ p.n/ p.F´.p//; if k.p/ 4´.p/ and n .mod ´.p// p.Ln/ C ¤ Á 2 I D 0; otherwise. With all of the above tools in hand, we now move to the proof of the theorem.

3. THEPROOFOF THEOREM 1 3.1. Item (i) By Lemma4 (d), we have

Lm Ln 5F m n F m n : D C2 2 ON THE ORDER OF APPEARANCE OF THE DIFFERENCE OF TWO LUCAS NUMBERS 645

Since F.m n/=2 divides Lm Ln, then .m n/=2 ´.Lm Ln/. Note that .m n/=2 and .m n/=2˙ are coprime which yields ˙ j C m2 n2 ´.Lm Ln/: (3.1) 4 j Also, F.m n/=2 F.m2 n2/=4 and since ˙ j gcd.F.m n/=2;F.m n/=2/ Fgcd..m n/=2;.m n/=2/ F1 1 C D C D D we conclude that F.m n/=2F.m n/=2 F.m2 n2/=4. Thus C j L L 5F m n F m n 5F m2 n2 F 2 2 Á; m n 2 m n D C2 j 4 j 5 4 where we used Lemma1 (c). Therefore Âm2 n2 Ã ´.Lm Ln/ 5 : (3.2) j 4 The combination between (3.1) and (3.2) yields m2 n2 Âm2 n2 Ã ´.Lm Ln/ ;5 : 2 4 4

Now, it sufﬁces to prove that 5F.m n/=2F.m n/=2 − F.m2 n2/=4. In fact, this follows because C 2 2 5.5F.m n/=2F.m n/=2/ 1 5..m n /=4/ C D C 2 2 > 5..m n /=4/ 5.F.m2 n2/=4/: D This completes the proof. 3.2. Item (ii) Proceeding as before, we can easily deduce that m2 n2 ´.Lm Ln/: (3.3) 4p j Here we used that gcd..m n/=2;.m n/=2/ p for all prime p gcd.m;n/. C D D 2 2 Now, we claim that Lm Ln divides F5Fp.m n /=4p, or equivalently, that 2 2 q.5F.m n/=2F.m n/=2/ q.F5Fp.m n /=4p/; C Ä holds for all primes q. Let us split the proof into four cases: Case 1. q 5. In this case, it follows directly that D 2 2 5.5F.m n/=2F.m n/=2/ 1 5..m n /=4/ C D C while 2 2 2 2 5.F5Fp.m n /=4p/ 1 5..m n /=4/ 5.Fp/ 5.p/ D C C 2 2 1 5..m n /=4/; D C 646 PAVEL TROJOVSKY´ where we used that 5.Fp/ 5.p/. D Case 2. q p 2;5. We have D ¤ p.5F.m n/=2F.m n/=2/ p..m ın/=2/ p.F´.p//; C D C C where ı 1;1 and we used that p cannot divide both F.m n/=2 and F.m n/=2, otherwise2´.p/ f wouldg divide p which is an absurdity by LemmaC 3. On the other hand, we have

2 2 p.F5Fp.m n /=4p/ p..m ın/=2/ p..m ın/=2/ 1 p.F´.p//: D C C C The result follows because p..m ın/=2/ 1, since p gcd.m;n/ > 2. D Case 3. q 2;5;p. In this case, we have ¤ 2 2 q.5F.m n/=2F.m n/=2/ q..m n /=4/ q.F´.q//; C D C 2 2 where is 1 or 2 according to the value of q..m n /=4/ is 1 or not, respectively. For the other valuation, we obtain 2 2 2 2 q.F5Fp.m n /=4p/ q.Fp/ q..m n /=4/ q.F´.q//: D C C If 1, the conclusion is immediate. For 2, then ´.q/ p and so ´.q/ p. D D j D Therefore q.Fp/ q.F / and the result follows. D ´.q/ Case 4. q 2. Since 2.5F.m n/=2F.m n/=2/ 2.F.m n/=2/ 2.F.m n/=2/, we need to considerD two subcases:C D C C Subcase 4.1 p 3. In this case, 3 divides only one among .m n/=2 and .m ¤ C n/=2. So, we have 2.5F.m n/=2F.m n/=2/ 1, when 3 .m n/=2 (because .m C D j C C n/=2 3 .mod 6/) and when 3 .m n/=2 one obtains 2.5F.m n/=2F.m n/=2/ 3 Á j C D or 2.m n/ 1 depending on .m n/=4 is odd or even, respectively. However, it is easy to infer thatC 8 1; if 3 .m n/=2 < j C I 2 2 − 2.F5Fp.m n /=4p/ 3; if 24 .m n/=2 D I : 2.m n/ 1; if 24 .m n/=2; C j yielding the desired inequality. Subcase 4.2 p 3. In this case, we get D 4; if .m n/=4 is odd 2.5F.m n/=2F.m n/=2/ I C D 2.m n/ 2; if .m n/=4 is even: C Now, note that 3 m n and 8 .m2 n2/ and then 12 5.m2 n2/=3. Hence, by Lemma5, j ˙ j j 2 2 2.F5.m2 n2/=3/ 2..m n /=3/ 2 2.m n/ 3 > 2.m n/ 2 4: D C D C C The result follows. ON THE ORDER OF APPEARANCE OF THE DIFFERENCE OF TWO LUCAS NUMBERS 647

2 2 Summarizing, we proved that Lm Ln divides F5Fp.m n /=4p and so, by Lemma 2 (c), we obtain 2 2 ´.Lm Lm/ 5Fp.m n /=4p: (3.4) j By (3.3) and (3.4), one concludes that 2 2 ´.Lm Lm/ t.m n /=4p; D where t divides 5Fp, that is, 5Fp st, for some positive integer s. Thus, in order to complete the proof of theorem,D it sufﬁces to prove that s 1. Suppose, to derive a contradiction, that s > 1 and let q be a prime factor of s.D Note that q p unless p 5. ¤ D p 5 (and then q 5). In this case, 25 ts and so t 1;5 or 25. Note that D D D D 2 2 2 2 5.5F.m n/=2F.m n/=2/ 1 5.m n / > 5.m n / 5.F.m2 n2/=4/: C D C D Thus 5F.m n/=2F.m n/=2 − F.m2 n2/=4 and the result follows. C p 5 (and so q 5). Since 5Fp ts, then q Fp yielding ´.q/ p. Therefore ¤ ¤ D j D q.5F.m n/=2F.m n/=2/ q..m ın/=2/ 2q.F´.q//; C D C C where we used that q divides both F.m n/=2 and F.m n/=2, here ı 0;1 . On the other hand, C 2 f g

q.5F.m n/=2F.m n/=2/ q.Fp/ q..m ın/=2/ q.s/ q.F´.q// C D C C C q..m ın/=2/ q.s/ 2q.F / D C C ´.q/ < q.5F.m n/=2F.m n/=2/; C since q s and we used that q.Fp/ q.F /. The proof of theorem is complete. j D ´.q/

4. ACKNOWLEDGEMENTS The author thanks for support to Speciﬁc Research Project of Faculty of Science, University of Hradec Kralove, No. 2101, 2017.

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[6] F. Luca and C. Pomerance, “On the local behavior of the order of appearance in the Fibonacci sequence.” Int. J. Number Theory, vol. 10, no. 4, pp. 915–933, 2014, doi: 10.1142/S1793042114500079. [7] D. Marques, “On the order of appearance of integers at most one away from Fibonacci numbers,” Fibonacci Quart., vol. 50, no. 1, pp. 36–43, 2012, doi: 10.1155/2011/407643. [8] D. Marques, “On integer numbers with locally smallest order of appearance in the Fibonacci sequence.” Int. J. Math. Math. Sci., vol. 2011, p. 4, 2011, doi: 10.1155/2011/407643. [9] D. Marques, “The order of appearance of powers of Fibonacci and Lucas numbers.” Fibonacci Q., vol. 50, no. 3, pp. 239–245, 2012. [10] D. Marques, “Sharper upper bounds for the order of appearance in the Fibonacci sequence.” Fibon- acci Q., vol. 51, no. 3, pp. 233–238, 2013. [11] P. Ribenboim, My numbers, my friends. Popular lectures on number theory. New York, NY: Springer, 2000. doi: 10.1007/b98892. [12] D. Robinson, “The Fibonacci matrix modulo m.” Fibonacci Q., vol. 1, no. 2, pp. 29–36, 1963. [13] H. Salle,´ “A maximum value for the rank of apparition of integers in recursive sequences.” Fibon- acci Q., vol. 13, pp. 159–161, 1975. [14] N. J. A. Sloane, “The On-Line Encyclopedia of Integer Sequences.” [Online]. Available: ttp://www.research.att.com/$ sim$njas/sequences/ n Author’s address

Pavel Trojovsky´ Faculty of Science University of Hradec Kralov´ e,´ Department of Mathematics, Rokitanskeho´ 62, Hradec Kralov´ e´ 50003, Czech Republic E-mail address: [email protected]