A FEW EQUIVALENCES OF WALL-SUN-SUN PRIME CONJECTURE
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1 International Journal of Mathematics & Alications Vol 4, No 1, (June 2011), A FEW EQUIVALENCES OF WALL-SUN-SUN PRIME CONJECTURE ARPAN SAHA AND KARTHIK C S ABSTRACT: In this aer, we rove a few lemmas concerning Fibonacci numbers modulo rimes and rovide a few statements that are equivalent to Wall-Sun-Sun Prime Conjecture Further, we investigate the conjecture through heuristic arguments and roose a coule of additional conjectures for future research 1 INTRODUCTION The Fibonacci sequence {F n } n 0 (defined as F 0 = 0, F 1 = 1 and F n = F n 1 + F n 2 for all n 2) has harbored great interest owing to its wide occurrence in combinatorial roblems such as that regarding the number of ways of tiling a 2 n rectangle with 2 1 dominoes and other numerous roerties it exhibits For instance, F m + n = F m 1 F n + F m F n + 1 and, as E Lucas had discovered, F gcd (m, n) = gcd(f m, F n ) Moreover, the work of D D Wall, Z H Sun and Z W Sun [3][4] regarding what has come to be known as the Wall-Sun-Sun Prime Conjecture, had demonstrated intimate links between the Fibonacci sequence and Fermat s Last Theorem [4] Though the latter ie Fermat s Last Theorem was roved in 199 by Andrew Wiles and Richard Taylor [][6], the Wall-Sun-Sun Prime Conjecture continues to generate interest This may be artly due to the fact that the Fibonacci sequence is interesting in its own right and artly due to the fact that it may lead to a relatively elementary aroach to Fermat s Last Theorem as comared to Wiles roof involving bijections between ellitic and modular forms The Wall-Sun-Sun Prime Conjecture is as follows: Statement 1: There does not exist a rime such that 2 F () where () is the Legendre symbol ie 1 if 1(mod ) 1 if 2(mod ) 0 if 0(mod ) (Such rimes shall henceforth be referred to as Wall-Sun-Sun rimes) We shall rovide a few statements equivalent to the above But we will first require a few definitions and results
2 78 Aran Saha and Karthik C S Definition: For a given ositive integer n, κ (n) is the least ositive integer m such that n F m Definition: For a given ositive integer n, π (n) is the least ositive integer m such that n F m and F m (mod n) This is often referred to as the Pisano eriod The existence of π(n) for any ositive integer n follows from the Pigeonhole Princile and the well-ordering of ositive integers [3]; the existence of κ (n) follows thence 2 BACKGROUND RESULTS We list here the results that we shall be using for demonstrating the equivalences discussed in the subsequent section Lemma 1: Let m and n be ositive integers We claim that n F m if and only if κ(n) m Proof: Both the necessity and sufficiency follow from the standard result due to E Lucas that gcd (F m, F k ) = F gcd (m, k) for any ositive integers m and k Here, if n F m then n gcd(f m, F κ(n) ) = F gcd (m, κ (n)) But F κ(n), by definition is the least Fibonacci number divisible by n (the Fibonacci numbers are an increasing sequence) So, κ (n) gcd (m, κ(n)) which imlies κ (n) = gcd (m, κ (n)) ie κ (n) m Conversely, if κ (n) m, then gcd (F m, F κ (n) ) = F gcd (m, κ (n)) = F κ (n) ie F κ (n) F m from which we conclude n F m Lemma 2: Let l be the highest ower of a ositive integer n dividing F κ(n) If n F m for some ositive integer m, then n l F m Proof: As, gcd(f m, F κ(n) ) = F gcd (m, κ (n)) = F κ(n) whenever n divides F m, we have F κ(n) F m Since n l F κ (n), we get n l F m as well Lemma 3: Let be a rime If F m and / F, then F m Proof: Let us recall Siebeck s formula for F mn n n j n j F mn = F j F m F m 1 j j 0 where m and n are ositive integers Let us ut n = We would then have j j F m = F j F m F m 1 j j 0 1 j j = F j F m F m 1 F F m j j 1
3 A Few Equivalences of Wall-Sun-Sun Prime Conjecture 79 On taking the equality modulo, we have F m F F m (mod ) as divides ( j ) for 1 j 1 Now from Fermat s Little Theorem we obtain, F m F F m (mod ) from which the result follows Lemma 4: Let l be the highest ower of a rime dividing F κ(n) We have, κ( l + 1 ) = κ() Proof: We ut n = and m = κ () in Siebeck s formula j j F κ() = F j F ()() F 1 j j 0 Let F κ() = l γ for some ositive integer γ not divisible by Then, lj j j F κ() = F j F () 1 j j 0 = F F F l 1 1 2( l 2) l j j j () 1() 1 j j 2 j As does not divide γ or F κ() 1 and we know that l 1, this imlies that the highest ower of dividing F κ() is l + 1 Now, by Lemma 1, we gather that κ( l + 1 ) κ() Let a be a ositive integer such that κ() = aκ( l + 1 ) Since l + 1 F, we have κ( l + 1) F κ( l + and again by Lemma 1, we have 1) κ() κ( l + 1 ) Let b be a ositive integer such that κ( l + 1 ) = bκ() So, κ() = abκ() which is to say, = ab Now a can be or 1 as is rime If a =, then we would have κ() = κ( l + 1 ) ie l + 1 F κ() This contradicts our assumtion, so we conclude a = 1 ie κ( l + 1 ) = κ() Lemma : For all ositive integers m and n, F mκ(n) + 1 F m κ(n) + 1 (mod n 2 ) Proof: We roceed by induction The base case m = 1 is trivial For the rest, we shall invoke the standard result, F r + s = F r F s F r 1 F s, r, s
4 80 Aran Saha and Karthik C S Let us assume that the congruence holds for all ositive integers m < κ, κ Then, F λκ(n) + 1 = F (λ 1) κ(n) κ(n) = F (λ 1) κ(n)+1 F κ(n) F (λ 1) κ(n) F κ(n) F λ 1 λκ(n) + 1 λκ(n) + 1 (mod n 2 ) (by inductive hyothesis) F λ λκ(n) + 1 (mod n 2 ) Hence, the congruence holds for m = λ as well The Lemma is thus roved Lemma 6: Let n be a ositive integer and let Ω n (z) denote the order of a ositive integer z modulo n We have, π(n) = κ(n) Ω n (F κ(n) + 1 ) Proof: The least ositive integer m for which, F mκ(n) (mod n) ie F m 1 (mod n) (by Lemma ) κ(n) + 1 is Ω n (F κ(n) + 1 ) Since gcd (F κ(n) + 1, F κ(n) ) = 1, we have that n, which divides F κ(n), is relatively rime to F κ(n) + 1 Hence, Ω n (F κ(n) + 1 ) is well-defined Now, from our definition of π(n), the Lemma immediately follows Lemma 7: Let r and n be ositive integers and be a rime Let α r and β r be residues modulo such that We claim that F rπ(n) α r (mod 2 ) F rπ(n) + 1 β r + 1 (mod 2 ) α r rα 1 (mod ) β r rβ 1 (mod ) Proof: We first note that the above notation is well-defined as, F rπ(n) 0 (mod ) F rπ(n) (mod ) We now roceed by induction The base case r = 1 is trivial For the rest, we first assume that the Lemma holds for r < From induction hyothesis, we have, F ( 1)π() α 1 ( 1) α 1 (mod 2 ) F ( 1)π() + 1 β ( 1) β (mod 2 )
5 A Few Equivalences of Wall-Sun-Sun Prime Conjecture 81 Now for the inductive stes: F π() F ( 1)π() + π() This gives us Similarly, F π() 1 F ( 1)π() + F π() F ( 1)π() + 1 ((β 1 α 1 ) + 1)( 1) α 1 + α 1 (( 1) β 1 + 1) α 1 (mod 2 ) α α 1 (mod ) This gives us F π() + 1 F ( 1)π() π() F π() F ( 1)π() + F π() + 1 F ( 1)π() + 1 ( 1) α (β 1 + 1) (( 1) β 1 + 1) β (mod 2 ) β β 1 (mod ) Thus the Lemma holds for all ositive integers r Lemma 8: Let be a rime We claim that π( 2 ) equals either π() or π() Proof: Firstly, we know that π(n) π(n 2 ) from Theorem stated in [3] and simle counting arguments Hence, π( 2 ) is of the form ξπ() where ξ is a ositive integer We continue with the notation α 1 and β 1 as introduced in Lemma 7 and investigate three cases: Case (i): α 1 0 (mod ) In this case, α 1, 2α 1, 3α 1, ( 1) α 1 are not congruent to zero modulo Thus, by Lemma 7, we have for all ositive integers ξ <, However, F ξπ() 0 (mod 2 ) α 1 0 (mod ) F π() 0 (mod 2 ) (by Lemma 7) F π() (mod 2 ) (by Lemma 7) Since π() is the least ositive integer g such that F g and F g + 1 are 0 and 1 modulo 2 resectively, we conclude π( 2 ) π() Although this Lemma is very well established, we hoe that we have given a new and more elementary roof
6 82 Aran Saha and Karthik C S Case (ii): β 1 0 (mod ) In this case, β 1, 2β 1, 3β 1, ( 1) β 1 are not congruent to zero modulo as well Thus, by Lemma 7, we have for all ositive integers ξ <, However, F ξπ() (mod 2 ) β 1 0 (mod ) F π() 0 (mod 2 ) (by Lemma 7) F π() (mod 2 ) (by Lemma 7) Since π() is the least ositive integer g such that F g and F g + 1 are 0 and 1 modulo 2 resectively, we conclude π( 2 ) π() Case (iii): α 1 β 1 0 (mod ) In this case, we directly have by definition F π() 0 (mod 2 ) F π() (mod 2 ) Hence, by again using the result π() π( 2 ), we have π( 2 ) = π() The Lemma is thus roved Consider the following statements: 3 A FEW EQUIVALENCES Statement 2: For any given rime, the highest ower of dividing F κ() is 1 Statement 3: Let m be a ositive integer and be a rime If 2 F m, then is a roer divisor of m Statement 4: For every rime, we have, π( 2 ) = π() With the above statements in mind, we have the following Theorem: Theorem: Statements 1, 2, 3 and 4 are equivalent Proof: We rove the bidirectional imlications: Statements 1 2, 2 3 and 2 4 To rove that Statement 1 imlies Statement 2, we use the result in [7], [8]: F ()
7 A Few Equivalences of Wall-Sun-Sun Prime Conjecture 83 According to Statement 1, 2 / F () 2 / () F, which by Lemma 2 leads us to conclude To rove that Statement 2 imlies Statement 1, we see that if the highest ower of dividing F κ() is 1 then, by Lemma 4, κ( 2 ) = κ() But, /() excet for the case =, which can be easily ruled out as is not a Wall-Sun-Sun rime Thus, we deduce / 2 () /() Using Lemma 1, we conclude that 2 F ie is not a Wall-Sun-Sun () rime Since, can be any arbitrary rime, we see that there does not exist any Wall-Sun-Sun rime To rove that Statement 2 imlies Statement 3, we see that if 1 is the highest ower of dividing F κ(), in accordance with Statement 2, then, by Lemma 4, κ( 2 ) = κ() Now, by Lemma 1, if 2 divides F m then κ( 2 ) ie κ() divides m In other words, is a roer divisor of m To rove that Statement 3 imlies Statement 2, we assume 2 F κ() By Statement 3, is a roer divisor of κ() So, we let κ() = c where c > 1 We have, by Lemma 3, F or F c But both and c are less than κ(), which according to definition is the least ositive integer n such that F n This is a contradiction Hence 2 / F () It follows that Statements 2 and 3 are equivalent To rove that Statement 2 imlies Statement 4, we use Lemma 4 in conjunction with Statement 2 to note κ( 2 ) = κ() But by Lemma, we know that: We have, as a consequence: F κ() + 1 F κ() + 1 (mod 2 ) Ω 2(F κ( 2 ) + 1 ) = Ω 2 (F κ() + 1 ) If we denote the quantity on either side as ω, then Hence, we would also have F ω 1 (mod κ () ) F ω 1 (mod ) κ () + 1 By Fermat s Little Theorem, we have F ω F ω κ() + 1 κ () (mod ) Though the equivalence between Statements 1 and 4 has been well established, we have rovided an elementary roof
8 84 Aran Saha and Karthik C S This means that Ω (F κ() + 1 ) ω ie Ω (F κ() + 1 ) Ω 2(F κ( ) whence we get, 2 ) + 1 But we have, Ω (F κ() + 1 ) Ω 2(F κ( 2 ) + 1 ) κ( 2 ) = κ() κ( 2 ) > κ() κ( 2 ) Ω 2 (F κ( 2 ) + 1 ) > κ() Ω (F κ() + 1 ) And as a consequence of Lemma 6 we have, Hence by Lemma 8, π() < π( 2 ) π( 2 ) = π() To rove that Statement 4 imlies Statement 2, we merely note that if every Fibonacci number divisible by a given rime was also divisible by 2, then π( 2 ) would be equal to π() Hence, Statement 4 imlies Statement 2 The equivalence is thus roved 4 HEURISTIC ARGUMENTS Firstly, some exciting results have been roved by A S Elsenhans and J Jahnel in [2] and we would request the readers to go through them An investigation regarding Wall-Sun-Sun Prime Conjecture carried out in [2] makes us believe that it might be true The oular version of the conjecture is its equivalent Statement 4 And from Lemma 8 we have that the conjecture imlies that there are no solutions to the equation below in rime numbers: π( 2 ) = π() However it would be interesting to find solutions to the above equation over all ositive integers Regarding this, we conjecture that the only solutions to the equation: π(n 2 ) = π(n), n are n = 6 and n = 12 Although no clear reason resents itself to us now, as to why the number 6 has such an interesting relationshi with its Pisano eriod, we can seculate why 12 follows it u The Pisano eriod function, π bears certain striking similarities to Euler s totient function, As indicated by comuter investigation, for instance, both seem to obey similar relations: φ( n ) = n 1 φ() [1] π( n ) = n 1 π() [2]
9 A Few Equivalences of Wall-Sun-Sun Prime Conjecture 8 for all rimes Further results such as: φ(mn) = (m)(n), if gcd (m, n) = 1, m, n [1] φ(mn) = l cm(π(m), π(n)), if gcd (m, n) = 1, m, n [2] confirm that there might be deeer links between the two functions Now considering the above equations, it is easier to areciate why π(6) = π(12) = π(6 2 ) = π(12 2 ) On a different note, it has been intuitively argued that π( 2 ) = π() for rime [2] So it is reasonable to exect every rime to divide π( 2 ) However for small values of n, it can be verified that: n π(n 2 ), n Keeing in mind Lemma 4 and certain results mentioned in [2], we claim that: n π(n 2 ), n If we see the above claim in the light of Lemma 6, we attain a better insight into the heart of the roblem, which only becomes more comelling when we bound π(n 2 ) by, π(n) π(n 2 ) nπ(n), n The roof of the inequality is omitted here, but we encourage the reader to rove them (Hint: Use Pigeonhole Princile) Note that no easily detectable attern emerges, as to when the equality holds for the uer bound Also, we have already conjectured regarding the condition when the equality holds for the lower bound Now assuming the above bounds on π(n 2 ), the claim that n π(n 2 ), n, becomes even more intriguing Summarizing, we conjecture the following statements: 1 The only solutions for the equation π(n 2 ) = π(n) over ositive integers are 6 and 12 2 n π(n 2 ), n ACKNOWLEDGMENTS We would like to thank Suryateja Gavva for heling us verify the roofs, checking for any lases of logic and roviding us with the much needed motivation to comlete this aer We would also like to thank Vihang Mehta for heling us cross-refer certain sources REFERENCES [1] T Aostol, (1998),Introduction to Analytic Number Theory, New York, Sringer, ISBN , 28 [2] A S Elsenhans, and J Jahnel, (2010), The Fibonacci sequence modulo 2 An Investigation by Comuter for < 10 14, arxiv: v1 [3] D D Wall, (1960), Fibonacci series modulo m, American Mathematical Monthly, 67, 2-32
10 86 Aran Saha and Karthik C S [4] Z W Sun, and Z H Sun, (1992), Fibonacci Numbers and Fermat s Last Theorem,Acta Mathematica, 60(4), [] A Wiles, (199), Modular Ellitic Curves and Fermat s Last Theorem, Annals of Mathematics, 141(3), [6] A Wiles, and R Taylor, (199), Ring Theoretical Proerties of Certain Hecke Algebras, Annals of Mathematics, 141(3), 3-72 [7] P Ribenboim, (1996), The New Book of Prime Number Records, New York, Sringer, ISBN: , 64 [8] F Lemmermeyer, (2000), Recirocity Laws, New York: Sringer, ISBN: , ex , Aran Saha * & Karthik C S ** Indian Institute of Technology (IIT), Bombay s: aran_saha@iitbacin * & karthikcs@iitbacin **
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