Journal of Computer Science & Computational Mathematics, Volume 3, Issue 4, December2013 57

The Fibonacci Primes Under Modulo 4

Qing Zou1 and Jun Steed Huang2

1 Department of Mathematics, Suqian College, Yangzhou University, 399 South Huanghe, Jiangsu 223800, P.R. China, 2 Suqian College, Jiangsu University, 399 South Huanghe, Jiangsu 223800, P.R. China, Corresponding addresses [email protected], [email protected]

Abstract: The research on prime numbers is an interesting topic in Analytic Number Theory. The Fibonacci sequence is one of the most The Fibonacci numbers are also closely related to Lucas popular sequences in mathematics. In this paper, we will discuss the numbers, since they are a complementary pair of Lucas prime numbers of the Fibonacci sequence, which we call Fibonacci sequences [5], [6]. primes. The goal of this paper is to prove that starting with , F5 = 5 all the Fibonacci primes p are satisfied with p º1 (mod 4), on There are many applications of the Fibonacci sequence, the basis of the Pythagorean theorem. including computer algorithms such as the Fibonacci heap [7] data structure, and graphs called Fibonacci cubes [8] used for Keywords: Fibonacci sequence; Fibonacci primes; Pythagorean interconnecting parallel and distributed systems. The triple; Algebraic Number Theory. Fibonacci numbers are also Nature's numbering system. They appear almost everywhere in Nature [9], from the leaf arrangement in plants, to the pattern of the florets of a flower, 1. Introduction the bracts of a pinecone, or the scales of a pineapple [10]. The Fibonacci numbers are therefore applicable to the growth of The Fibonacci sequence (sometimes called Fibonacci every living thing, including a single cell, a grain of wheat, a numbers), which is named after Leonardo Fibonacci (c.1170 – hive of bees, and even mankind [10]. c.1250), are the numbers in the following integer sequence: 1,1,2,3,5,8,13,21,34,55,89,144, In light of the applications mentioned above, it is clear that the In some cases [1], the first two numbers in the Fibonacci Fibonacci sequence plays an important role not only in sequence are 0 and 1. It depends on the chosen starting point mathematics but also in our daily lives, such as the of the sequence. In this paper, we choose the first two obfuscation or encryption of the software code in C++ [11]. numbers to be 1 and 1, since it relates immediately to the As such, research on the sequence becomes more and more rabbit family size. important. Fibonacci primes are one of such topics.

The constitution of the sequence is such that each subsequent A Fibonacci prime is a Fibonacci number that is prime. The number is the sum of the previous two. In mathematics terms, few seeds are: the sequence of the Fibonacci numbers is defined by the Fn F3 = 2,F4 = 3,F5 = 5,F7 =13,F11 = 89, recurrence formula F = 233,F =1597, 13 17 Fn = Fn-1 + Fn-2 Although Fibonacci primes with thousands of digits have with initial values [2] been found, it is not known whether there are infinitely many Fibonacci prime numbers [12]. It is one of the unsolved F1 =1,F2 =1. problems in mathematics. The divisibility of Fibonacci There are some interesting identities on Fibonacci numbers. numbers by a prime, p, is related to the Legendre symbol For example, since æ p ö é ù é ùn ç ÷. If p is a prime number then Fn+1 Fn 1 1 è 5 ø ê ú=ê ú , æ ö ëê Fn Fn-1 ûú ë 1 0 û p Fp º ç ÷ (mod p), we have è 5 ø 2 n Fn+1Fn-1 - Fn = (-1) . F æ pö º 0 (mod p). p-ç ÷ Also, there exists è 5 ø 2 2 2 2 It is also not known whether there exists a prime, , such Fn+3 = 2Fn+2 +2Fn+1 - Fn . that, The common item equation of the Fibonacci sequence is 2 F æ pö º 0 (mod p ). n n p-ç ÷ éæ ö æ ö ù è 5 ø 5 ê 1+ 5 1- 5 ú Fn = ç ÷ -ç ÷ . 5 ëêè 2 ø è 2 ø ûú Such primes (if there are any) would be called Wall-Sun-Sun primes or Fibonacci-Wieferich primes. Wall–Sun–Sun primes Several methods can be used to get this equation [3]. It is are named after Donald Dines Wall, Zhi Hong Sun and Zhi known that the Fibonacci sequence is closely related to golden Wei Sun; Fibonacci-Wieferich primes are named after Arthur ratio [4]. Wieferich. 58 Journal of Computer Science & Computational Mathematics, Volume 3, Issue 4, December2013

2 2 2 2 m n m -n 2mn m +n It is not known whether there are an infinite number of 3 4 Fibonacci primes, but we can determine the property of F3 = 2 F2 =1 F5 = 5 Fibonacci primes, which is the goal of this paper. 5 12 F4 = 3 F3 = 2 F7 =13

2. Revisiting the Pythagorean Triple 16 30 F5 = 5 F4 = 3 F9 = 34 A Pythagorean triple is made of three positive integers, x, y and , such that 2 2 2. Such a triple is often denoted as 39 80 z x + y = z F6 = 8 F5 = 5 F11 = 89 (x,y,z), and a well-known example found a few thousands years ago is (3,4,5). The name is due to the Pythagorean theorem (Chinese: Shang Gao Theorem), describing that any 2 2 Fi+1 Fi F - F 2FiFi-1 F2i+1 given right triangle has side lengths that satisfy the formula i+1 i 2 2 2 2 . So, Pythagorean triples describe the relationship = Fi+2Fi-1 = F2i+2 - Fi+1 = Fi + Fi+1 of the three integer side lengths of a right triangle.

Euclid's formula [13] is fundamental for generating According to the table listed above, we can see Pythagorean triples when given an arbitrary pair of positive F2 = (2FF )2 +(F2 - F2 )2. integers m and n with m > n . The formula states that 2i+1 i i-1 i+1 i That is to say, starting with , every second Fibonacci integers number can be the largest number in a Pythagorean triple. □ ì 2 2 ï x = m - n í y = 2mn Lemma 1 tells us that starting with , the Fibonacci number with an odd index is the length of a right triangle with ï 2 2 î z = m + n integer sides.

form a Pythagorean triple. The triple generated by Euclid's Lemma 2. Starting with , Fibonacci primes have a formula is primitive if, and only if, and are coprime prime index. In other words, starting with , all and m-n is odd. Fibonacci primes have an odd index.

Euclid's formula does not generate all triples, but can generate Proof. Suppose there exists a Fibonacci prime with a all primitive triples. This can be remedied by inserting an composite index m and a is a factor of m , then a divides additional parameter, k , into the formula. The following m . For if a divides m , then F divides F , meaning that formula will generate all Pythagorean triples uniquely: a m is a factor of . It is a contradiction. Therefore, lemma 2 Fa Fm is proved. □ ì 2 2 ï x = k(m - n ) ï What we need to point out is that though starting with , í y = k(2mn) Fibonacci primes have a prime index, not every prime larger ï 2 2 îï z = k(m + n ) than 5 is an example of a Fibonacci prime, F = 4181 = 37 ´113 is a counter example. 19

where m,n and are positive integers with , odd, and with and coprime. 4. The Modulo Theorem and Proof The Euclidian formula is so prominent that the proof can be By lemma 1 and lemma 2, we can easily get that, starting with seen in almost every elementary number theory book [14]. So, , all the Fibonacci primes can be the length of the we will not give the proof here. hypotenuse of a right triangle with integer sides.

Consequently, if we want to prove that, starting with , 3. The Parity Lemmas and Proofs all the Fibonacci primes satisfy p º1(mod 4), we just need to prove that for all prime numbers p, if p º1 mod 4 , then The Pythagorean triples can be used to prove the following ( ) lemma. x2 + y2 = p2 has integer solutions.

Lemma 1. Starting with F5 = 5 , every second Fibonacci Before we prove the proposition, we need to introduce another number is the length of the hypotenuse of a right triangle with conclusion in Algebraic Number Theory. integer sides, or in other words, the largest number in a Pythagorean triple. Lemma 3. (1) If is a prime number congruent to 1 modulo 4, then there exists a prime element a in [i] such that Proof. We can prove this lemma by the following table. p =aa (a is also a prime element of ). Journal of Computer Science & Computational Mathematics, Volume 3, Issue 4, December2013 59

Theorem 1. Starting with F5 = 5 , all Fibonacci primes (2) If p is a prime number congruent to 3 modulo 4, then satisfy p º1(mod 4). is a prime element in [i].

Proof. Let's prove this conclusion in two steps. Note. An element a in is a prime element if it satisfies: Step 1. Let be a prime number and , then

(a) is nonzero and not a unit (invertible element). there exists a right triangle with integer sides such that the (b) If a,b Î [i] and ab Îa [i], then a Îa [i] or length of hypotenuse is .

b Îa [i]. Because , by lemma 3(1), can be written Proof. (1) Let be a prime number and p º1(mod 4), since as p =aa , where is a prime element in . Suppose ì 2 æ ö p-1 ï 1, p º1 (mod 4), a = x+iy,(x,y Î ), we have -1 2 ç ÷ = (-1) = í p2 =a2a 2 = x2 + y2. è p ø ï -1, p º 3 (mod 4), î If we show x ¹ 0,y ¹ 0, we see that is the length of the æ-1ö æ a ö hypotenuse of a right triangle whose three sides are | x |,| y | then ç ÷ =1 , where ç ÷ represents Legendre symbol. è p ø è p ø and . If x = 0 or y = 0, the argument of is a multiple 2 p Also, there exists an integer a such that a º -1(mod p). of , and thus there exists an integer m such that Since 4 2 a = mb, where b Î {1,1+i,i,-1,-1+i}. (a+i)(a-i) = a +1Î p [i], a+i Î/ p [i], a-i Î/ p [i]. So, is not a prime element in . This contradicts the uniqueness of the prime factorization in 2 2 2 . It is obvious that the equation 2 = x + y does not On the other hand, because is not a unit in , it follows have a solution satisfying x ¹ 0,y ¹ 0. from the prime factorization of in that there exists a prime element in dividing . Let Step 2. If is a prime number and p º 3(mod 4) , then p =ab,b Î [i]. Because is not a prime element, b there will be no right triangle that the length of the hypotenuse is . is not a unit, we have 2 p =ab ×ab =aa ×bb, 2 2 2 Let p = x + y (x,y Î ). If we set a = x +iy, then we and because both aa and bb are natural numbers, 2 have p =aa . By lemma 3(2), is a prime element in 2 2 must be one of the divisors of p , i.e. 1, p, p . If aa =1, . It follows from the uniqueness of factorization that then would be a unit and we will get a contradiction. If

2 a = p´(±1,or ±i). aa = p , then would be a unit, since bb =1. Thus, This implies x = 0 or y = 0. there must be p =aa . According to the two steps shown above, lemma 1 and lemma (2) Let be a prime number and p º 3(mod 4), and let 2, the theorem is proved. □

be a prime element in dividing p. Let . Then, as is shown above, we have 5. Conclusion 2 p =ab ×ab =aa ×bb. In this paper, we have studied the Fibonacci numbers that are 2 2 prime and started with the simple seed 1 and 1. We have Because and we cannot write p = x + y aa ¹1 proven that, in this case, modulo 4 be 1 can be used to offer a ( x, y Î ) (since there exists x, y Î satisfying quick judgment and if the number obtained during the computer program iteration belongs to a prime. This operation if and only if p º1 mod 4 or p = 2 ), we can ( ) saves the complicated computation time for finding a prime get that p ¹aa . Thus, there must be aa = p2,bb =1, that fits well for applications like obfuscation of intellectual property protected code. The applications of this new finding and is a unit. Hence is a prime element. □ p =ab will not be limited.

Note. Moreover, we can prove that in the former part of this 6. Acknowledgement lemma, a [i] is not equal to a [i]. This work was partially supported by GenieView subsidiary By this lemma, we have M2M Technology 331.7.49 Project, and the Suqian Excellence Expert Allowance 2013.18.14.5.9.2. 60 Journal of Computer Science & Computational Mathematics, Volume 3, Issue 4, December2013

[5] Leonid Bedratyuk, "Derivations and Identitites for We are grateful to following individuals for their Fibonacci and Lucas Polynomials", The Fibonacci immeasurable helps and contributions to this research work: Quarterly, vol.51,pp.351-366, 2013. Jiangyong Gu, Runping Ye, from our university. [6] Mohamed Taoufiq Damir, Bernadette Faye, Florian Luca, Amadou Tall, "Members of Lucas Sequences Whose Euler Function Is a Power of 2", The Fibonacci 6.1 Author Contributions Quarterly, vol.52,pp.3-9, 2014. The manuscript was written through contributions of all [7] Brodal G. S. L., Lagogiannis G., Tarjan R. E., "Strict authors. All authors have given approval to the final version Fibonacci Heaps", Proceedings of the 44th Symposium of the manuscript. Qing Zou conceived and designed the on Theory of Computing - STOC '12, 2012, pp. 1177. original work that led to this submission, and played an [8] Dedó, Ernesto, Torri, Damiano, Salvi, Norma Zagaglia, important role in proof of the results. Jun S. Huang improved "The observability of the Fibonacci and the Lucas the manuscript, collected some references and discussed the cubes", Discrete Mathematics, vol.255(1-3),pp. 55-63, applications with Qing Zou during the process of the writing. 2002. [9] Douady S., Couder Y., "Phyllotaxis as a Dynamical Self 6.2 Conflict of Interests Organizing Process". Journal of Theoretical Biology, vol.178 ,pp. 255-74, 1996. The authors declare no Conflicts of Interest. [10] S. L. Basin, "The Fibonacci sequence as it appears in

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