ABSTRACT 1. INTRODUCTION

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1 ON PIMES AN TEMS OF PIME O INEX IN THE LEHME SEUENCES John H. Jaroma epartment of Mathematical Sciences, Loyola College in Maryland, Baltimore, M 110 Submitted May 004-Final evision March 006 ABSTACT It is nown with a very small number of exceptions, for a term of a Lehmer sequence {U n, } to be prime its index must be prime. For example, F 4 U 4 1, 1 3 is prime. Also, U n 1, is prime for n 6, 8, 9, 10, 15, 5, 5, 65, while V n 1, is prime for n 9, 1, and 0. This criterion extends to the companion Lehmer sequences {V n, }, with the exception primality may occur if the index is a power of two. Furthermore, given an arbitrary prime p or any positive integer, there does not exist an explicit means for determining whether U p, V p, or V is prime. In 000, V. robot provided conditions under which if p and p 1 are prime then F p is composite. A short while later, L. Somer considered primes of the form p ± 1, as well as generalized robot s theorem to the Lucas sequences. Most recently, J. Jaroma extended Somer s findings to the companion Lucas sequences. In this paper, we shall generalize all of the aforementioned results from the Lucas sequences to the Lehmer sequences. 1. INTOUCTION In [1], V. robot introduced the following theorem. It gave a set of sufficient conditions for a Fibonacci number of prime index to be composite. Theorem 1 robot: Let p > 7 be a prime satisfying the following two conditions: 1. p mod 5 or p 4 mod 5. p 1 is prime Then, F p is composite. In fact, p 1 F p. In [6], L. Somer generalized the above theorem to the Lucas sequences. Let P and be nonzero relatively prime integers. The Lucas sequences {U n P, } are defined as U n+ P U n+1 U n, U 0 0, U 1 1, n {0, 1,... }. 1 The companion Lucas sequences {V n P, } are given by V n+ P V n+1 V n, V 0, V 1 P, n {0, 1,... }. Furthermore, if we let P 4 denote the discriminant of the characteristic equation of 1 and, then Somer s extension of Theorem 1 may be stated as Theorem Somer: Let {UP, } be a Lucas sequence and p be an odd prime such p ± If p 1 is a prime, 1 and 1, then p 1 U p.. If p + 1 is a prime, 1, then p + 1 U p. Somer s result was originally given in [6, pg. 435] in terms of the second-order linear recurrence satisfying u n+ au n+1 + bu n, u 0 0, u 1 1, and a, b Z. It was also noted in 0

2 ON PIMES AN TEMS OF PIME O INEX IN THE LEHME SEUENCES [] in light of the second line of Table 1 on pg. 373 of [5], if the hypotheses of the above theorem are strengthened to include the conditions p ± 1 P and gcdp, 1, then Theorem can be reformulated to provide necessary and sufficient conditions for p ± 1 to be prime. Also, found in [] are the following results for the companion Lucas sequences. Theorem 3: Let {V P, } be a companion Lucas sequence and let p be an odd prime such p ± 1 P. 1. Let 1. Then, p 1 V p if and only if p 1 is prime.. Let 1 and 1. Then, p + 1 V p if and only if p + 1 is prime.. THE ESULTS We shall now extend all of the aforementioned results to the larger family of Lehmer sequences. To this end, let p be of the arbitrary form p α p α 1 ± 1, 3 where, α 1, and for 1 i, α i {0, 1,... } and the p i are distinct odd primes. Any odd p may always be described in either of the two forms described by 3. Now, letting and be any pair of relatively prime integers, the Lehmer sequences {U n, } are recursively defined as U n+, U n+1 U n, U 0 0, U 1 1, n {0, 1,... }. 4 Also, the companion Lehmer sequences {V n, } are similarly given by V n+, V n+1 V n, V 0, V 1, n {0, 1,... }. 5 As Lehmer had declared in [3], we say m divides when and only when m. Let 4 be the discriminant of the characteristic equation of 4 and 5, the following Legendre symbols will be used in this paper. σ p, τ p, ɛ Finally, the ran of apparition of a number N is the index of the first term of the underlying sequence contains N as a factor. We shall let ωp represent the ran of apparition of p in {U n, } and λp denote the ran of apparition of p in {V n, }. Our forthcoming generalization will require the following lemmata found in [3]. Lemma 1: GCU n, V n 1 or. p. Lemma : If N ± 1 is the ran of apparition of N, then N is prime. Lemma 3: If p then p U pσɛ. Lemma 4: Let p. Then, p U pσɛ if and only if σ τ. Lemma 5: If the ran of apparition of p, ωp, is odd then p V n, for any value of n. If ωp is even, say, then p V for all n and no other term of the sequence may contain p as a factor. Lemma 6: U n is divisible by p if and only if n ωp. 03

3 ON PIMES AN TEMS OF PIME O INEX IN THE LEHME SEUENCES We now establish our results. Theorem 4: Let {U n, } be a Lehmer sequence, α 1, for 1 i assume α i 0, p α p α 1 1 is prime, and gcdp + 1, 1. For the following Jacobi symbols, let either 1 or 1. If p + 1 α+1 p α 1 1 is prime, then α+1 p α 1 1 U α p α 1 Proof: Let p + 1 α+1 p α 1 1 be prime. Now, σɛ Hence, from Lemma 3, α+1 p α 1 1 U α+1 p α 1 by Lemma 4, α+1 p α 1 1 U α p α 1 pα 1. pα pα Furthermore, since σ τ, Theorem 5: Let {U n, } be a Lehmer sequence, α 1, for 1 i assume α i 0, p α p α is prime, and gcdp 1, 1. For the following Jacobi symbols, let 1 or 1. If p 1 α+1 p α is prime, then α+1 p α U α p α 1 pα +1. Proof: Let p 1 α+1 p α be prime. Since σɛ 1, it follows α+1 p α 1 +1 U α+1 p α 1 pα +. Finally, as σ τ, we have α+1 p α 1 +1 U α p α 1 pα +1. Let N q β 1 1 qβ qβ r r be any odd composite number where q i < q j whenever i < j, gcdn, 1, and U NεN 0 mod N, where εn denotes the Jacobi symbol, N. Then, N is a Lucas pseudoprime for {U n P, }. Furthermore, N is called a Lucas d-pseudoprime provided there exists a Lucas sequence {U n P, } satisfying gcdn, P 1, where N is a pseudoprime for {U n P, } and ωn NεN d is the ran of apparition of N in {U n P, }. Moreover, we say n is a Lehmer d-pseudoprime if there exists a Lehmer sequence {U n, } satisfying gcdn, 1: ωn NσNɛN d, where σn and ɛn denote the Jacobi symbols /N and /N, respectively. It follows from results found in [3] and from Chapter 5 in [4] an odd composite integer N is a Lehmer d-pseudoprime if and only if it is a Lucas d-pseudoprime. Hence, If we consider the wor of Somer, who in [5] showed for any fixed d: 4 d, there exists only a finite number of Lucas d-pseudoprimes, then the statements of Theorems 4 and 5 are able to be strengthened to necessary and sufficient ones. For this purpose, we note if d, then the only Lucas and hence, Lehmer -pseudoprime is 3. This occurs, for instance, in {U4, 1}. emar 1: In [5], Somer illustrates N 9 is the only composite number for which ωn N±1. Therefore, as previously noted, N 9 is the only Lehmer -pseudoprime. Furthermore, the ran of apparition of 9 is always equal to 4 and never equal to a prime value. This follows since N 9 is a square, and so, σn ɛn 1 whenever gcd9, 1. Thus, for any Lehmer sequence in which 9 is a Lehmer -pseudoprime, we have ω9 9σ9ɛ Therefore, all other N with this particular ran of apparition must be prime. We now restate Theorems 4 and 5 to reflect necessary and sufficient conditions for the primality of p ± 1 α+1 p α 1 ± 1. The demonstration we give for Theorem 6 is for necessity only. The sufficiency portion follows from Theorem 4. The proof of Theorem 7 is similar and omitted. 04

4 ON PIMES AN TEMS OF PIME O INEX IN THE LEHME SEUENCES Theorem 6: Let {U n, } be a Lehmer sequence, α 1, for 1 i assume α i 0, p α p α 1 1 is prime, and gcdp + 1, 1. Let 1 or 1. Then, p + 1 α+1 p α 1 1 is prime if and only if α+1 p α 1 1 U α p α 1 pα 1. Proof: In each case, let α+1 p α 1 1 U α p α 1 of U α p α 1 pα pα 1. Since the index 1 is prime and equal to α+1 p α 1 pα 11, we have ω α+1 p α 1 1 α p α 1 1. Therefore, by emar 1, α+1 p α 1 1 is prime. Theorem 7: Let {U n, } be a Lehmer sequence, α 1, for 1 i assume α i 0, p α p α 1 +1 is prime, and gcd, 1. Let 1 or 1. Then, p 1 α+1 p α is prime if and only if α+1 p α U α p α 1 pα +1. Theorems 6 and 7 will now be extended to the companion Lehmer sequences after offering a comment on a connection between Lehmer s original proof of the Lucas-Lehmer test given in [3] and establishing a primality test for numbers of the form n ± 1 using the companion Lehmer sequences, in general. emar : The Lucas-Lehmer test states n 1 is prime if and only if n 1 divides the n 1st term of the sequence, 4, 14, 194, 37634,..., S,..., where, S S1. Equivalently, the terms of the described sequence are those of the companion Lehmer sequence {V n, 1} with indices equal to. Hence, we may restate the said result as, n 1 is prime if and only if n 1 V n1, 1. Moreover, based on the proof of the result given in [3], we may also infer if n for some 1, then n ± 1 is prime if and only if n ± 1 V n, when n±1 n±1 1. Finally, when n and n ± 1 is a prime, then n ± 1 is either a Mersenne prime for the case n 1 or a Fermat prime for the case n + 1. Theorem 8: Let {V n, } be a companion Lehmer sequence, α 0, for 1 i assume α i 0, n α p α 1 1, and gcdn + 1, 1. Let 1 or If n is a prime, then n + 1 α+1 p α 1 1 is prime if and only if α+1 p α 1 1 V α p α 1 pα 1.. If n α, then n + 1 α is prime if and only if α V α. Proof: 1. As σɛ 1, by Lemma 3, α+1 p α 1 1 U α+1 p α 1 pα. Also, since σ τ, then α+1 p α 1 1 U α p α 1 pα 1. Since α+1 p α 1 α p α 1 1, by Lemma 6, either ωα+1 p α 1 1 or ωα+1 p α 1. Now, α+1 p α 1 1. So, ω α+1 p α 1, and by Lemma 5, 1 α+1 p α 1 1. Thus, ω α+1 p α 1 1 α+1 p α 1 05

5 ON PIMES AN TEMS OF PIME O INEX IN THE LEHME SEUENCES λ α+1 p α 1 1 α p α 1 V α p α 1 pα 1. Now, let α+1 p α 1 1 V α p α 1 1. Therefore, α+1 p α 1 1 p α 1. Since U n U n V n, it follows α+1 p α 1 1 U α+1 p α 1 p α. By Lemma 1, α+1 p α 1 1 U α p α 1 p α 1. So, as ωα+1 p α 1 1, we have ωα+1 p α 1 1 α+1 p α 1. Therefore, by Lemma, α+1 p α 1 1 is prime.. Let m + 1 α be prime. As σɛ 1, by Lemma 3, α U α+1. Since στ 1; is, σ τ, we have α U α. Thus, ω α α+1 and λ α α. Therefore, α V α. On the other hand, if α V α, then by the identity U n U n V n, it follows α U α+1. However, by Lemma 1, α U α. So, ω α α+1. Therefore, by Lemma, α is prime. Theorem 9: Let {V n, } be a companion Lehmer sequence, α 0, for 1 i assume α i 0, n α p α 1 + 1, and gcdn + 1, 1. Let 1 or If n is a prime, then n 1 α+1 p α is prime if and only if α+1 p α V α p α 1 pα +1.. If n α, then n 1 α+1 1 is prime if and only if α+1 1 V α. 3. A ANK OF APPAITION INTEPETATION We say p has maximal ran of apparition in {U n, } provided ωp p±1. If p divides the term, say U q, where q is a prime, then we may necessarily conclude ωp q. As a result, Theorems 4 through 9 are easily restated in order each may provide a ran of apparition result. Theorems 8A and 9A provide maximal ran of apparition results. Theorem 4A: Let {U n, } be a Lehmer sequence, α 1, for 1 i assume α i 0, p α p α 1 1 is prime, and gcdp + 1, 1. Let either 1 or 1. If α+1 p α 1 1 is prime, then ω α+1 p α 1 1 α p α 1 1 and λ α+1 p α 1 1 does not exist. Theorem 5A: Let {U n, } be a Lehmer sequence, α 1, for 1 i assume α i 0, p α p α 1 +1 is prime and gcd, 1. Let 1 or 1. If α+1 p α is prime, then ω α+1 p α α p α and λα+1 p α does not exist. Theorem 6A: Let {U n, } be a Lehmer sequence, α 1, for 1 i assume α i 0, p α p α 1 1 is prime, and gcdp + 1, 1. Let 1 or 1. Then, α+1 p α 1 1 is prime ω α+1 p α 1 1 α p α 1 exist. 1 and λα+1 p α 1 1 does not 06

6 ON PIMES AN TEMS OF PIME O INEX IN THE LEHME SEUENCES Theorem 7A: Let {U n, } be a Lehmer sequence, α 1, for 1 i assume α i 0, p α p α 1 +1 is prime, and gcd, 1. Let 1 or 1. Then, α+1 p α is prime ω α+1 p α α p α 1 exist. + 1 and λα+1 p α does not Theorem 8A: Let {V n, } be a companion Lehmer sequence, α 0, for 1 i assume α i 0, n α p α 1 1, and gcdn + 1, 1. Also, let 1 or If n is a prime, then n + 1 α+1 p α 1 1 is prime ωα+1 p α 1 p α 1 α+1 p α 1 and λα p α 1 1 α+1 p α If n α, then α+1 +1 is prime ω α+1 +1 α+1 and λ α+1 +1 α. Theorem 9A: Let {V n, } be a companion Lehmer sequence, α 0, for 1 i assume α i 0, n α p α 1 + 1, and gcdn + 1, 1. Also, let 1 or If n is a prime, then n 1 α+1 p α is prime + 1 α+1 p α 1 + and λ α+1 p α If n α, then α+1 1 is prime ω α+1 1 α+1 and λ α+1 1 ω α+1 p α 1 α p α 1 α. ACKNOWLEGMENT The author would lie to than an anonymous referee whose careful reading of the manuscript, insight, and expertise led to many improvements in this paper, among which included the strengthening of Theorems 8, 8A, 9, 9A to the case of V n when n and the extension of Theorems 6, 6A, 7, 7A to the Lehmer sequences. EFEENCES [1] V. robot. On Primes in the Fibonacci Sequence. Fibonacci uarterly : [] J. H. Jaroma. Extension of a Theorem of Somer to the Companion Lucas Sequences. Proceedings of the 11th Conference of Fibonacci Numbers and their Applications. Kluwer, The Netherlands, 006, pp [3]. H. Lehmer. An Extended Theory of Lucas Functions. Ann. Math., Second Series : [4] L. Somer. The ivisibility and Modular Properties of th-order Linear ecurrences Over the ing of Integers of an Algebraic Number Field With espect to Prime Ideals. Ph.. issertation, The University of Illinois at Urbana-Champaign, [5] L. Somer. On Lucas d-pseudoprimes. Fibonacci Numbers and Their Applications. 7: Ed. G. E. Bergum et al. ordrecht: Kluwer Academic Publishers,

7 ON PIMES AN TEMS OF PIME O INEX IN THE LEHME SEUENCES [6] L. Somer. Generalization of a Theorem of robot. Fibonacci uarterly 40 00: AMS Classification Numbers: 11A51, 11B39 08