ON DIVISORS OF LUCAS AND LEHMER NUMBERS

This aer submitte to Acta Mathematica. ON DIVISORS OF LUCAS AND LEHMER NUMBERS C.L. STEWART 1. Introuction Let u n be the n-th term of a Lucas sequence or a Lehmer sequence. In this article we shall establish an estimate from below for the greatest rime factor of u n which is of the form n exlog n/104 log log n). In so oing we are able to resolve a question of Schinzel from 1962 an a conjecture of Erős from 1965. In aition we are able to give the first general imrovement on results of Bang from 1886 an Carmichael from 1912. Let α an β be comlex numbers such that α + β an αβ are non-zero corime integers an α/β is not a root of unity. Put u n = α n β n )/α β) for n 0. The integers u n are known as Lucas numbers an their ivisibility roerties have been stuie by Euler, Lagrange, Gauss, Dirichlet an others see [11, Chater XVII]). In 1876 Lucas [24] announce several new results concerning Lucas sequences u n ) n=0 an in a substantial aer in 1878 [25] he gave a systematic treatment of the ivisibility roerties of Lucas numbers an inicate some of the contexts in which they aeare. Much later Matijasevic [26] aeale to these roerties in his solution of Hilbert s 10th roblem. For any integer m let P m) enote the greatest rime factor of m with the convention that P m) = 1 when m is 1, 0 or 1. In 1912 Carmichael [8] rove that if α an β are real an n > 12 then P u n ) n 1. 1) Results of this character ha been establishe earlier for integers of the form a n b n where a an b are integers with a > b > 0. Inee Zsigmony [49] in 1892 an Birkhoff an Vaniver [6] in 1904 rove that for n > 2 P a n b n ) n + 1, 2) while in the secial case that b = 1 the result is ue to Bang [4] in 1886. In 1930 Lehmer [22] showe that the ivisibility roerties of Lucas numbers hol in a more general setting. Suose that α + β) 2 an αβ are corime non-zero integers with α/β not a root of unity an, for n > 0, ut { α n β n )/α β) for n o, ũ n = α n β n )/α 2 β 2 ) for n even. Research suorte in art by the Canaa Research Chairs Program an by Grant A3528 from the Natural Sciences an Engineering Research Council of Canaa. 1

2 C.L. STEWART Integers of the above form have come to be known as Lehmer numbers. Observe that Lucas numbers are also Lehmer numbers u to a multilicative factor of α + β when n is even. In 1955 War [45] rove that if α an β are real then for n > 18, P ũ n ) n 1, 3) an four years later Durst [13] observe that 3) hols for n > 12. A rime number is sai to be a rimitive ivisor of a Lucas number u n if ivies u n but oes not ivie α β) 2 u 2 u n 1. Similarly is sai to be a rimitive ivisor of a Lehmer number ũ n if ivies ũ n but oes not ivie α 2 β 2 ) 2 ũ 3 ũ n 1. For any integer n > 0 an any air of comlex numbers α an β, we enote the n-th cyclotomic olynomial in α an β by Φ n α, β), so n Φ n α, β) = α ζ j β), j=1 j,n)=1 where ζ is a rimitive n-th root of unity. One may check, see [38], that Φ n α, β) is an integer for n > 2 if α + β) 2 an αβ are integers. Further, see Lemma 6 of [38], if, in aition, α + β) 2 an αβ are corime non-zero integers, α/β is not a root of unity an n > 4 an n is not 6 or 12 then P n/3, n)) ivies Φ n α, β) to at most the first ower an all other rime factors of Φ n α, β) are congruent to 1 or 1 moulo n. The last assertion can be strengthene in the case that α an β are corime integers to the assertion that all other rime factors of Φ n α, β) are congruent to 1 mo n). Since α n β n = n Φ α, β), 4) Φ 1 α, β) = α β an Φ 2 α, β) = α + β we see that if n excees 2 an is a rimitive ivisor of a Lucas number u n or Lehmer number ũ n then ivies Φ n α, β). Further, a rimitive ivisor of a Lucas number u n or Lehmer number ũ n is not a ivisor of n an so it is congruent to ±1 mo n). Estimates 1), 2) an 3) follow as consequences of the fact that the n-th term of the sequences in question ossesses a rimitive ivisor. It was not until 1962 that this aroach was extene to the case where α an β are not real by Schinzel [31]. He rove, by means of an estimate for linear forms in two logarithms of algebraic numbers ue to Gelfon [17], that there is a ositive number C, which is effectively comutable in terms of α an β, such that if n excees C then ũ n ossesses a rimitive ivisor. In 1974 Schinzel [35] emloye an estimate of Baker [2] for linear forms in the logarithms of algebraic numbers to show that C can be relace by a ositive number C 0, which oes not een on α an β, an in 1977 Stewart [39] showe C 0 coul be taken to be e 452 4 67. This was subsequently refine by Voutier [42, 43] to 30030. In aition Stewart [39] rove that C 0 can be taken to be 6 for Lucas numbers an 12 for Lehmer numbers with finitely many excetions an that the excetions coul be etermine by solving a finite number of Thue equations. This rogram was successfully carrie out by Bilu, Hanrot an Voutier [5] an as a consequence they were able to show that for n > 30 the n-th term of a Lucas or Lehmer sequence has a rimitive ivisor. Thus 1) an 3) hol for n > 30 without the restriction that α an β be real. In 1962 Schinzel [30] aske if there exists a air of integers a, b with ab ifferent from ±2c 2 an ±c h with h 2 for which P a n b n ) excees 2n for all sufficiently

ON DIVISORS OF LUCAS AND LEHMER NUMBERS 3 large n. In 1965 Erős [14] conjecture that P 2 n 1) as n. n Thirty five years later Murty an Wong [28] showe that Erős conjecture is a consequence of the abc conjecture [41]. They rove, subject to the abc conjecture, that if ε is a ositive real number an a an b are integers with a > b > 0 then P a n b n ) > n 2 ε, rovie that n is sufficiently large in terms of a, b an ε. In 2004 Murata an Pomerance [27] rove, subject to the Generalize Riemann Hyothesis, that P 2 n 1) > n 4/3 / log log n 5) for a set of ositive integers n of asymtotic ensity 1. The first unconitional refinement of 2) was obtaine by Schinzel [30] in 1962. He rove that if a an b are corime an ab is a square or twice a square then P a n b n ) 2n + 1 rovie that one exclues the cases n = 4, 6, 12 when a = 2 an b = 1. Schinzel rove his result by showing that the term a n b n was ivisible by at least 2 rimitive ivisors. To rove this result he aeale to an Aurifeuillian factorization of Φ n. Rotkiewicz [29] extene Schinzel s argument to treat Lucas numbers an then Schinzel [32, 33, 34] in a sequence of articles gave conitions uner which Lehmer numbers ossess at least 2 rimitive ivisors an so uner which 3) hols with n + 1 in lace of n 1, see also [21]. In 1975 Stewart [37] rove that if κ is a ositive real number with κ < 1/ log 2 then P a n b n )/n tens to infinity with n rovie that n runs through those integers with at most κ log log n istinct rime factors, see also [15]. Stewart [38] in the case that α an β are real an Shorey an Stewart [36] in the case that α an β are not real generalize this work to Lucas an Lehmer sequences. Let α an β be comlex numbers such that α + β) 2 an αβ are non-zero relatively rime integers with α/β not a root of unity. For any ositive integer n let ωn) enote the number of istinct rime factors of n an ut qn) = 2 ωn), the number of square-free ivisors of n. Further let ϕn) be the number of ositive integers less than or equal to n an corime with n. They showe, recall 4), if n > 3) has at most κ log log n istinct rime factors then P Φ n α, β)) > Cϕn) log n)/qn), 6) where C is a ositive number which is effectively comutable in terms of α, β an κ only. The roofs een on lower bouns for linear forms in the logarithms of algebraic numbers in the comlex case when α an β are real an in the -aic case otherwise. The urose of the resent aer is to answer in the affirmative the question ose by Schinzel [30] an to rove Erős conjecture in the wier context of Lucas an Lehmer numbers. Theorem 1.1. Let α an β be comlex numbers such that α + β) 2 an αβ are non-zero integers an α/β is not a root of unity. There exists a ositive number C, which is effectively comutable in terms of ωαβ) an the iscriminant of Qα/β), such that for n > C, P Φ n α, β)) > n exlog n/104 log log n). 7)

4 C.L. STEWART Our result, with the ai of 4) gives an imrovement of 1), 2), 3) an 6), answers the question of Schinzel an roves the conjecture of Erős. Secifically, if a an b are integers with a > b > 0 then P a n b n ) > n exlog n/104 log log n), 8) for n sufficiently large in terms of the number of istinct rime factors of ab. We remark that the factor 104 which occurs on the right han sie of 7) has no arithmetical significance. Instea it is etermine by the current quality of the estimates for linear forms in -aic logarithms of algebraic numbers. In fact we coul relace 104 by any number strictly larger than 14e 2. The roof eens uon estimates for linear forms in the logarithms of algebraic numbers in the comlex an the -aic case. In articular it eens uon a result of Yu [48] where imrovements uon the eenence on the arameter in the lower bouns for linear forms in -aic logarithms of algebraic numbers are establishe. This allows us to estimate irectly the orer of rimes iviing Φ n α, β). The estimates are non-trivial for small rimes an, coule with an estimate from below for Φ n α, β), they allow us to show that we must have a large rime ivisor of Φ n α, β) since otherwise the total non-archimeean contribution from the rimes oes not balance that of Φ n α, β). By contrast for the roof of 6) a much weaker assumtion on the greatest rime factor is imose an it leas to the conclusion that then Φ n α, β) is ivisible by many small rimes. This art of the argument from [36] an [38] was also emloye in Murata an Pomerance s [27] roof of 5) an in estimates of Stewart [40] for the greatest square-free factor of ũ n. My initial roof of the conjecture of Erős utilize an estimate for linear forms in -aic logarithms establishe by Yu [47]. In orer to treat also Lucas an Lehmer numbers however, I nee the more refine estimate obtaine in [48], see 3. For any non-zero integer x let or x enote the -aic orer of x. Our next result follows from a secial case of Lemma 8 of this aer. Lemma 8 yiels a crucial ste in the roof of Theorem 1. An unusual feature of the roof of Lemma 8 is that we artificially inflate the number of terms which occur in the -aic linear form in logarithms which aears in the argument. We have chosen to highlight it in the integer case. Theorem 1.2. Let a an b be integers with a > b > 0. There exists a number C 1, which is effectively comutable in terms of ωab), such that if is a rime number which oes not ivie ab an which excees C 1 an n is an integer with n 2 then or a n b n ) < ex log /52 log log ) log a + or n. 9) If a an b are integers with a > b > 0, n is an integer with n 2 an is an o rime number which oes not ivie ab an excees C 1 then or a 1 b 1 ) < ex log /52 log log ) log a. Yamaa [46], by making use of a refinement of an estimate of Bugeau an Laurent [7] for linear forms in two -aic logarithms, rove that there is a ositive number C 2, which is effectively comutable in terms of ωa), such that or a 1 1) < C 2 /log ) 2 ) log a. 10)

ON DIVISORS OF LUCAS AND LEHMER NUMBERS 5 By following our roof of Theorem 1 an using 10) in lace of Lemma 8 it is ossible to show that there exist ositive numbers C 3, C 4 an C 5, which are effectively comutable in terms of ωa) such that if n excees C 3 then P a n 1) > C 4 ϕn)log n log log n) 1 2 an so, by Theorem 328 of [19], P a n 1) > C 5 nlog n/ log log n) 1 2. 11) This gives an alternative roof of the conjecture of Erős, although the lower boun 11) is weaker than the boun 8). The research for this aer was one in art uring visits to the Hong Kong University of Science an Technology, Institut es Hautes Étues Scientifiques an the Erwin Schröinger International Institute for Mathematical Physics an I woul like to exress my gratitue to these institutions for their hositality. In aition I wish to thank Professor Kunrui Yu for helful remarks concerning the resentation of this article an for our extensive iscussions on estimates for linear forms in -aic logarithms which le to [48]. 2. Preliminary lemmas Let α an β be comlex numbers such that α + β) 2 an αβ are non-zero integers an α/β is not a root of unity. We shall assume, without loss of generality, that α β. Observe that r + s r s α =, β = 2 2 where r an s are non-zero integers with r s. Further Qα/β) = Q rs). Note that α 2 β 2 ) 2 = rs an we may write rs in the form m 2 with m a ositive integer an a square-free integer so that Q rs) = Q ). For any algebraic number γ let hγ) enote the absolute logarithmic height of γ. In articular if a 0 x γ 1 ) x γ ) in Z[x] is the minimal olynomial of γ over Z then hγ) = 1 log a 0 + log max1, γ j ). Notice that j=1 αβx α/β)x β/α) = αβx 2 α 2 + β 2 )x + αβ = αβx 2 α + β) 2 2αβ)x + αβ is a olynomial with integer coefficients an so either α/β is rational or the olynomial is a multile of the minimal olynomial of α/β. Therefore we have hα/β) log α. 12) We first recor a result escribing the rime factors of Φ n α, β). Lemma 2.1. Suose that α + β) 2 an αβ are corime. If n > 4 an n 6, 12 then P n/3, n)) ivies Φ n α, β) to at most the first ower. All other rime factors of Φ n α, β) are congruent to ±1 mo n). Proof. This is Lemma 6 of [38].

6 C.L. STEWART Let K be a finite extension of Q an let be a rime ieal in the ring of algebraic integers O K of K. Let O consist of 0 an the non-zero elements α of K for which has a non-negative exonent in the canonical ecomosition of the fractional ieal generate by α into rime ieals. Then let P be the unique rime ieal of O an ut K = O /P. Further for any α in O we let α be the image of α uner the resiue class ma that sens α to α + P in K. Our next result is motivate by work of Lucas [25] an Lehmer [22]. Let be an o rime an be an integer corime with. Recall that the Legenre symbol ) is 1 if is a quaratic resiue moulo an 1 otherwise. Lemma 2.2. Let be a square-free integer ifferent from 1, θ be an algebraic integer of egree 2 over Q in Q ) an let θ enote the algebraic conjugate of θ over Q. Suose that is a rime which oes not ivie 2θθ. Let be a rime ieal of the ring of algebraic integers of Q ) lying above. The orer of θ/θ in Q ) ) is a ivisor of 2 if ivies θ 2 θ 2 ) 2 an a ivisor of otherwise. Proof. We first note that θ an θ are -aic units. If ivies θ 2 θ 2 ) 2 then either ivies θ θ ) 2 or ivies θ + θ an in both cases θ/θ ) 2 1 mo ). Thus the orer of θ/θ ivies 2. Thus we may suose that oes not ivie 2θθ θ 2 θ 2 ) 2 an, in articular,. Since 2θ = θ + θ ) + θ θ ) an 2θ = θ + θ ) θ θ ) 13) we see, on raising both sies of the above equations to the -th ower an subtracting, that 2 θ θ ) 2θ θ ) is θ θ ) times an algebraic integer. Hence, since is o, θ θ θ θ θ θ ) 1 mo ). But θ θ ) 1 = θ θ ) 2 ) 1 θ θ ) 2 ) 2 mo ) an θ θ ) 2 ) ) =, so θ θ ) θ θ mo ). 14) By raising both sies of equation 13) to the -th ower an aing we fin that an since If θ+θ ) 2 ) = 1, θ + θ θ + θ θ + θ ) 1 mo ) ) = 1 then aing 14) an 15) we fin that θ + θ θ + θ 1 mo ). 15) 2 θ+1 θ +1 θ 2 θ 2 0 mo ) )

ON DIVISORS OF LUCAS AND LEHMER NUMBERS 7 hence, since oes not ivie 2θθ θ 2 θ 2 ) 2, an the result follows. If θ/θ ) +1 1 mo ) ) = 1 then subtracting 14) an 15) we fin that 2θθ θ 1 θ 1 θ 2 θ 2 0 mo ) hence, since oes not ivie 2θθ θ 2 θ 2 ) 2, an this comletes the roof. θ/θ ) 1 1 mo ) Let l an n be integers with n 1 an for each real number x let πx, n, l) enote the number of rimes not greater than x an congruent to l moulo n. We require a version of the Brun-Titchmarsh theorem, see [18, Theorem 3.8]. Lemma 2.3. If 1 n < x an n, l) = 1 then πx, n, l) < 3x/ϕn) logx/n)). Our next result gives an estimate for the rimes below a given boun which occur as the norm of an algebraic integer in the ring of algebraic integers of Qα/β). Lemma 2.4. Let be a squarefree integer with 1 an let k enote the k-th smallest rime of the form Nπ k = k where N enotes the norm from Q ) to Q an π k is an algebraic integer in Q ). Let ε be a ositive real number. There is a ositive number C, which is effectively comutable in terms of ε an, such that if k excees C then log k < 1 + ε) log k. Proof. Let K = Q ) an enote the ring of algebraic integers of K by O K. A rime is the norm of an element π of O K rovie that it is reresentable as the value of the rimitive quaratic form q K x, y) given by x 2 y 2 if 1 mo 4) an x 2 + xy + ) 1 4 y 2 if 1 mo 4). By [16, Chater VII, 2.14)], a rime is reresente by q K x, y) if an only if is not inert in K an the rime ieals of O K above have trivial narrow class in the narrow ieal class grou of K. Let K H be the strict Hilbert class fiel of K. Since K H is normal over K an G, the Galois grou of K H over K, is isomorhic with the narrow ieal class grou of K it follows that G = h +, the strict ieal class number of K, see Theorem 7.1.2 of [10]. The rime ieals of O K which o not ramify in K H an which are rincial are the only rime ieals of O K which o not ramify in K H an which slit comletely in K H, see Theorem 7.1.3 of [10].[ These ] rime ieals may be counte by the Chebotarev Density Theorem. Let KH /K enote the conjugacy class of Frobenius automorhisms corresoning to rime ieals P of O KH above. In articular, for each conjugacy class C of G we efine π C x, K H /K) to be the carinality] of the set of rime ieals of O K which are unramifie in K H, for which = C an for which N K/Q x. Denote by C 0 the conjugacy class [ KH /K consisting of the ientity element of G. Note that the number of inert rimes of O K for which N K/Q x is at most x 1/2. Thus the number of rimes u to x for which is the norm of an element π of O K is boune from below by π C0 x, K H /K) x 1/2. 16)

8 C.L. STEWART It follows from Theorem 1.3 an 1.4 of [23] that there is a ositive number C 1, which is effectively comutable in terms of, such that for x greater than C 1 the quantity 16) excees x 2h + log x. Further when x is at least 4h + k log k an x 2h + log x > k Thus, rovie 17) hols an x excees C 1, k/ log k > 4h +. 17) k < 4h + k log k. 18) Our result now follows from 18) on taking logarithms. 3. Estimates for linear forms in -aic logarithms of algebraic numbers Let α 1,..., α n be non-zero algebraic numbers an ut K = Qα 1,..., α n ) an = [K : Q]. Let be a rime ieal of the ring O K of algebraic integers in K lying above the rime number. Denote by e the ramification inex of an by f the resiue class egree of. For α in K with α 0 let or α be the exonent to which ivies the rincial fractional ieal generate by α in K an ut or 0 =. For any ositive integer m let ζ m = e 2πi/m an ut α 0 = ζ 2 u where ζ 2 u K an ζ 2 u+1 K. Suose that α 1,..., α n are multilicatively ineenent -aic units in K. Let α 0, α 1,..., α n be the images of α 0, α 1,..., α n resectively uner the resiue class ma at from the ring of -aic integers in K onto the resiue class fiel K at. For any set X let X enote its carinality. Let α 0, α 1,..., α n be the subgrou of K ) generate by α 0, α 1,..., α n. We efine δ by an if Denote log maxx, e) by log x. δ = 1 if [Kα 1/2 0, α 1/2 1,..., α 1/2 n ) : K] < 2 n+1 δ = f 1)/ α 0, α 1,..., α n [Kα 1/2 0, α 1/2 1,..., α 1/2 n ) : K] = 2 n+1. 19) Lemma 3.1. Let be a rime with 5 an let be an unramifie rime ieal of O K lying above. Let α 1,..., α n be multilicatively ineenent -aic units. Let b 1,..., b n be integers, not all zero, an ut Then B = max2, b 1,..., b n ). or α b1 1 αbn n 1) < Chα 1 ) hα n ) maxlog B, n + 1)5.4n + log ))

ON DIVISORS OF LUCAS AND LEHMER NUMBERS 9 where C = 376n + 1) 1/2 7e 1 2 f n max δ f log ) n n+2 log loge 4 n + 1)) ) n, e n f log ). Proof. We aly the Main Theorem of [48] an in 1.16) we take C 1 n,,, a)h 1) in lace of the minimum. Further 1.15) hols since our result is symmetric in the b i s. Next we note that, since is unramifie an 5, we may take c 1) = 1794, a 1) = 7 1 2, a1) 0 = 2 + log 7 an a 1) 1 = a 1) 2 = 5.25. We remark that conition 19) ensures that we may take {θ 1,..., θ n } to be {α 1,..., α n }. Finally the exlicit version of Dobrowolski s Theorem ue to Voutier [44] allows us to relace the first term in the maximum efining h 1) by log B. Therefore we fin that or α b1 1 αbn n 1) < C 1 hα 1 ) hα n ) maxlog B, G 1, n + 1)f log ) where G 1 = n + 1)2 + log 7)n + 5.25 + log2 + log 7)n + 5.25) + log ) an, on enoting log maxx, e) by log x, )) n 1 n + 1) n+1 n+2 log C 1 = 1794 7 2 n! 2 u f log ) 2 f ) n n max, e n f log ) maxloge 4 n + 1)), f log ). δ f log Note that 2 u 2 an f log log 5. Further, by Stirling s formula, see 6.1.38 of [1], n + 1) n+1 en+1 n + 1) 1/2 n! 2π an so where ) log B or α b1 1 αbn n 1) < C 2 hα 1 ) hα n ) max log 5, G 1 log 5, n + 1 C 2 = 1794 2 e n + 1) 1/2 7e 1 2π 2 f max δ n f log ) n n+2 log ) n ) loge, e n 4 n + 1)) f log. log 5 We next observe that G 1 n + 1)5.4n + log ) an as a consequence ) ) log B max log 5, G 1 log B log 5, n + 1 n + 1)5.4n + log ) = max,. 22) log 5 log 5 The result now follows from 20), 21) an 22). The key new feature in Yu s Main Theorem in [48],as comare with his estimate in [47], is the introuction of the factor δ. It is the resence of δ in the statement of Lemma 3.1 that allows us to exten our argument to the case when Qα/β) is ifferent from Q. 20) 21)

10 C.L. STEWART 4. Further reliminaries Let α + β) 2 an αβ be non-zero integers with α/β not a root of unity. We may suose that α β. Since there is a ositive number c 0 which excees 1 such that α c 0 we euce from Lemma 3 of [39], see also Lemmas 1 an 2 of [35], that there is a ositive number c 1 which we may suose excees log c 0 ) 1 such that for n > 0 log 2 + n log α log α n β n n c 1 logn + 1)) log α. 23) The roof of 23) eens uon an estimate for a linear form in the logarithms of two algebraic numbers ue to Baker [2]. For any ositive integer n let µn) enote the Möbius function of n. It follows from 4) that Φ n α, β) = nα n/ β n/ ) µ). 24) We may now euce, following the aroach of [35] an [39], our next result. Lemma 4.1. There exists an effectively comutable ositive number c such that if n > 2 then α ϕn) cqn) log n Φ n α, β) α ϕn)+cqn) log n, 25) where qn) = 2 ωn). Proof. By 24) log Φ n α, β) = µ) log α n/ β n/ n an so by 23) log Φ nα, β) µ) n log α n c 1 logn + 1) log α n µ) 0 since c 1 excees log c 0 ) 1. Our result now follows. Lemma 4.2. There exists an effectively comutable ositive number c 2 such that if n excees c 1 then log Φ n α, β) ϕn) log α. 26) 2 Proof. For n sufficiently large ϕn) > n/2 log log n an qn) < n 1/ log log n. Since α c 0 > 1 it follows from 25) that if n is sufficiently large as require. Φ n α, β) > α ϕn)/2, Lemma 4.3. Let n be an integer larger than 1, let be a rime which oes not ivie αβ an let be a rime ieal of the ring of algebraic integers of Qα/β) lying above which oes not ramify. Then there exists a ositive number C, which is effectively comutable in terms of ωαβ) an the iscriminant of Qα/β), such that if excees C then or α/β) n 1) < ex log /51.9 log log ) log α log n.

ON DIVISORS OF LUCAS AND LEHMER NUMBERS 11 Proof. Let c 3, c 4,... enote ositive numbers which are effectively comutable in terms of ωαβ) an the iscriminant of Qα/β). We remark that since α/β is of egree at most 2 over Q the iscriminant of Qα/β) etermines the fiel Qα/β) an so knowing it one may comute the class number an regulator of Qα/β) as well as the strict Hilbert class fiel of Qα/β) an the iscriminant of this fiel. Further let be a rime which oes not ivie 6αβ where is efine as in the first aragrah of 2. Put K = Qα/β) an { i if i K α 0 = 1 otherwise. Let v be the largest integer for which α/β = α j 0 θ2v, 27) with 0 j 3 an θ in K. To see that there is a largest such integer we first note that either there is a rime ieal q of O K, the ring of algebraic integers of K, lying above a rime q which occurs to a ositive exonent in the rincial fractional ieal generate by α/β or α/β is a unit. In the former case hα/β) 2 v 1 log q an in the latter case, since α/β is not a root of unity, there is a ositive number c 3, see [12], such that hα/β) 2 v c 3. Notice from 27) that hα/β) = 2 v hθ). 28) Further, by Kummer theory, see Lemma 3 of [3], [Kα 1/2 0, θ 1/2 ) : K] = 4. 29) Furthermore since αβ an α an β are algebraic integers or α/β) n 1) or α/β) 4n 1). 30) For any real number x let [x] enote the greatest integer less than or equal to x. Put [ ] log k =. 31) 51.8 log log Then, for > c 4, we fin that k 2 an ) k ) k k k max, e k log ) =. 32) log log Our roof slits into two cases. We shall first suose that Qα/β) = Q so that α an β are integers. For any ositive integer j with j 2 let j enote the j 1-th smallest rime which oes not ivie αβ. We ut an m = n2 v+2 33) α 1 = θ/ 2 k. Then ) m θ m θ 1 = m 2 m k 1 = α1 m m 2 m k 1 34) 2 k an by 27), 30), 33) an 34) or α/β) n 1) or α m 1 m 2 m k 1). 35)

12 C.L. STEWART Note that α 1, 2,..., k are multilicatively ineenent since α/β is not a root of unity an 2,..., k are rimes which o not ivie αβ. Further, since 2,..., k are ifferent from an oes not ivie αβ, we see that α 1, 2,..., k are -aic units. We now aly Lemma 3.1 with δ = 1, = 1, f = 1 an n = k to conclue that or α1 m m 2 m k 1) c 5 k + 1) 3 log 7e 1 ) k Put max k log t = ωαβ). Let q i enote the i-th rime number. Note that an thus k q k+t+1 2 ) ) k, e k log 2 + + log k k 1) log q k+t+1. By the rime number theorem with error term, for k > c 6, log m)hα 1 ) log 2 log k. 36) log 2 + + log k 1.001k 1) log k. 37) By the arithmetic geometric mean inequality log 2 + + log k log 2 log k k 1 an so, by 37), ) k 1 log 2 log k 1.001 log k) k 1. 38) Since hα 1 ) hθ) + log 2 k it follows from 37) that hα 1 ) c 7 hθ)k log k. 39) Further m = 2 v+2 n is at most n 2v+2 an so by 12) an 28) hθ) log m 4hα/β) log n 4 log α log n. 40) Thus, by 32), 35), 36), 38), 39) an 40), or α/β) n 1) < c 8 k 4 log 7e 1 ) k log k 1.001k log α log n. 2 log Therefore, by 31), for > c 9 log or α/β) n 1) < e 51.9 log log log α log n. 41) We now suose that [Qα/β) : Q] = 2. Let π 2,..., π k be elements of O K with the roerty that Nπ i ) = i where N enotes the norm from K to Q an where i is the i 1) th smallest rational rime number of this form which oes not ivie 2αβ. We now ut θ i = π i /π i where π i enotes the algebraic conjugate of π i in Qα/β). Notice that oes not ivie π i π i = i an if oes not ivie π i π i )2 then πi π i ) ) )2 =,

ON DIVISORS OF LUCAS AND LEHMER NUMBERS 13 since Qα/β) = Q ) = Qπ i ). Thus, by Lemma 2.2, the orer ) of θ i in Qα/β) ) is a ivisor of 2 if ivies πi 2 π 2 i )2 an a ivisor of otherwise. Since is o an oes ) not ivie we conclue that the orer of θ i in Qα/β) ) is a ivisor of. Put α 1 = θ/θ 2 θ k. 42) Then ) m θ m θ 1 = θ2 m θk m 1 θ 2 θ k an, by 27), 30), 33) an 42), or α/β) n 1) or α m 1 θ m 2 θ m k 1). 43) Observe that α 1, θ 2,..., θ k are multilicatively ineenent since α/β is not a root of unity, 2,..., k are rimes which o not ivie αβ an the rincial rime ieals [π i ] for i = 2,..., k o not ramify since 2. Further since 2,..., k are ifferent from an oes not ivie αβ we see that α 1, θ 2,..., θ k are -aic units. Notice that Kα 1/2 0, θ 1/2, θ 1/2 2,..., θ 1/2 k ) = Kα 1/2 0, α 1/2 1, θ 1/2 2,..., θ 1/2 k ). Further [Kα 1/2 0, θ 1/2, θ 1/2 2,..., θ 1/2 k ) : K] = 2 k+1, 44) since otherwise, by 29) an Kummer theory, see Lemma 3 of [3], there is an integer i with 2 i k an integers j 0,..., j i 1 with 0 j b 1 for b = 0,..., i 1 an an element γ of K for which θ i = α j0 0 θj1 θ j2 2 θji 1 i 1 γ2. 45) But the orer of the rime ieal [π i ] on the left-han sie of 45) is 1 whereas the orer on the right-han sie of 45) is even which is a contraiction. Thus 44) hols. Since oes not ivie the iscriminant of K an [K : Q] = 2 either slits, in ) which case f = 1 an = 1, or is inert, in which case f = 2 an ) see [20]. Observe that if = 1 then ) = 1, f /δ. ) 46) Let us now etermine α 0, θ, θ 2,..., θ k in the case = 1. By our earlier remarks the orer of θ i is a ivisor of + 1 for i = 2,..., k. Further by 27) since Nα/β) = 1 we fin that Nθ) = ±1 an so Nθ 2 ) = 1. By Hilbert s Theorem 90, see eg. Theorem 14.35 of [9], θ 2 = ϱ/ϱ where ϱ an ϱ are conjugate algebraic integers in Qα/β). Note that we may suose that the rincial ieals [ϱ] an [ϱ ] have no rincial ieal ivisors in common. Further since oes not ivie αβ an, since ) = 1, [] is a rincial rime ieal of O K an we note that oes not ivie ϱϱ. It follows from Lemma 2.2 that the orer of θ 2 in Qα/β) ) is a ivisor of + 1 hence θ has orer a ivisor of 2 + 1). Since α 4 0 = 1 we conclue that α 0, θ, θ 2,..., θ k 2 + 1)

14 C.L. STEWART an so δ = 2 1)/ α 0, θ, θ 2,..., θ k 1)/2. 47) We now aly Lemma 3.1 noting, by 46) an 47), that f /δ 2 2 / 1). Thus, by 32), or α1 m θ2 m θk m 1) c 10 k + 1) 3 log 7e 1 ) k ) k k 2 k 2 log log m)hα 1 )hθ 2 ) hθ k ). Notice that θ i = π i /π i an that ix π i /π i )x π i /π i) = i x 2 πi 2 +π 2 i )x+ i is the minimal olynomial of θ i over the integers since [π i ] is unramifie. Either the iscriminant of Qα/β) is negative in which case π i = π i or it is ositive in which case there is a funamental unit ε > 1 in O K. We may relace π i by π i ε u for any integer u an so without loss of generality we may suose that 1/2 i π i 1/2 i ε an consequently that 1/2 i ε 1 π i 1/2 i. Therefore an Let us ut Then hθ i ) 1 2 log iε 2 = 1 2 log i + log ε for > 0 hθ i ) 1 2 log i for < 0. R = { log ε for > 0 0 for < 0. for i = 2,..., k. In a similar fashion we fin that an so 48) hθ i ) 1 2 log i + R 49) hθ 2 θ k ) 1 2 log 2 k + R, 50) hα 1 ) hθ) + 1 2 log 2 k + R. 51) Put t 1 = ω2αβ). Let q i enote the i-th rime number which is reresentable as the norm of an element of O K. Note that k q k+t1 an thus log 2 + + log k k 1) log q k+t1. Therefore by Lemma 2.4 for k > c 11, an so, as for the roof of 38), Accoringly, since k k, for k > c 12 log 2 + + log k 1.0005k 1) log k 52) log 2 log k 1.0005 log k) k 1. 2 k 1 hθ 2 ) hθ k ) log 2 + 2R) log k + 2R) 1.001 log k) k 1. 53)

ON DIVISORS OF LUCAS AND LEHMER NUMBERS 15 Furthermore as for the roof of 39) an 40) we fin that from 51), an, from 12), 28) an 33), hα 1 ) c 13 hθ)k log k 54) hθ) log m 8 log α log n. 55) Thus by 43), 48), 51), 53), 54) an 55), ) 1 or α/β) n 1) < c 14 k 4 log 7e 1.001 k log k ) k log α log n. 56) 2 log Therefore, by 31), for > c 15 we again obtain 41) an the result follows. 5. Proof of Theorem 1.1 Let c 1, c 2,... enote ositive numbers which are effectively comutable in terms of ωαβ) an the iscriminant of Qα/β). Let g be the greatest common ivisor of α + β) 2 an αβ. Note that ϕn) is even for n > 2 an that Φ n α, β) = g ϕn)/2 Φ n α 1, β 1 ) where α 1 = α/ g an β 1 = β/ g. Further α 1 + β 1 ) 2 an α 1 β 1 are corime an lainly P Φ n α, β)) P Φ n α 1, β 1 )). Therefore we may assume, without loss of generality, that α + β) 2 an αβ are corime non-zero integers. By Lemma 4.2 there exists c 1 such that if n excees c 1 then On the other han Φ n α, β) = log Φ n α, β) ϕn) 2 Φ nα,β) log α. 57) orφnα,β). 58) If ivies Φ n α, β) then, by 4), oes not ivie αβ an so or Φ n α, β) or α/β) n 1), 59) where is a rime ieal of O K lying above. By Lemma 2.1 if ivies Φ n α, β) an is not P n/3, n)) then is at least n 1 an thus for n > c 2, by Lemma 4.3, Put or α/β) n 1) < ex log /51.9 log log ) log α log n. 60) P n = P Φ n α, β)). Then, by 58) an Lemma 2.1, log Φ n α, β) log n + log or Φ n α, β). 61) P n n Comaring 57) an 61) an using 59) an 60) we fin that, for n > c 3, ϕn) log α < c 4 log ) ex log /51.9 log log ) log α log n P n n

16 C.L. STEWART hence ϕn) log n < πp n, n, 1) + πp n, n, 1))P n ex log P n /51.95 log log P n ), an, by Lemma 2.3, ϕn) c 5 log n < Pn 2 ϕn) logp n /n) ex log P n/51.95 log log P n ). Since ϕn) > c 6 n/ log log n, for n > c 7, as require. P n > n exlog n/104 log log n), Since oes not ivie ab 6. Proof of Theorem 1.2 or a n b n ) = or a/g) n b/g) n ) where g is the greatest common ivisor of a an b. Thus we may assume, without loss of generality, that a an b are corime. Put u n = a n b n for n = 1, 2, an let l = l) be the smallest ositive integer for which ivies u l. Certainly ivies u 1. Further, as in the roof of Lemma 3 of [38], if n an m are ositive integers then u n, u m ) = u n,m). Thus if ivies u n then ivies u n,l). By the minimality of l we see that n, l) = l so that l ivies n. In articular l ivies 1. Furthermore, by 4), we see that or u l = or Φ l a, b). If l ivies n then, by Lemma 2 of [38], an so u n /u l, u l ) ivies n/l, 62) or u 1 = or u l 63) Suose that ivies Φ n a, b). Then ivies u n an so l ivies n. Put n = tl k with t, ) = 1 an k a non-negative integer. Since Φ n a, b) ivies u n /u n/t for t > 1 we see from 62), since t, ) = 1, that t = 1. Thus n = l k. For any ositive integer m ) u m /u m = b m 1) + b m 2) u m + + u 1 m 2 an if is not 2 an ivies u m then or u m /u m ) = 1. It then follows that if is an o rime then or Φ l ka, b) = 1 for k = 1, 2,.

ON DIVISORS OF LUCAS AND LEHMER NUMBERS 17 If n is a ositive integer not ivisible by l = l) then u n = 1. On the other han if l ivies n an is o then u n = u l n/l. 64) It now follows from 63) an 64) an the fact that l 1 that if is an o rime an l ivies n then u n = u 1 n. 65) Therefore if is an o rime an n a ositive integer or a n b n ) or a 1 b 1 ) + or n, 66) an our result now follows from 66) on taking n = 1 in Lemma 8. References [1] ABRAMOWITZ,M. & STEGUN,I.A., Hanbook of Mathematical Functions with Formulas, Grahs an Mathematical Tables, Dover Publications, New York, 1972. [2] BAKER,A., A sharening of the bouns for linear forms in logarithms, Acta Arith. 21 1972), 117 129. [3] BAKER,A. & STARK,H., On a funamental inequality in number theory, Ann. Math. 94 1971), 190 199. [4] BANG,A.S., Taltheoretiske unersøgelser, Tisskrift for Mat. 4 1886), 70 78, 130 137. [5] BILU,Y., HANROT,G. & VOUTIER,P.M., Existence of rimitive ivisors of Lucas an Lehmer numbers, J. reine angew. Math. 539 2001), 75 122. [6] BIRKHOFF,G.D.& VANDIVER,H.S., On the integral ivisors of a n b n, Ann. Math. 5 1904), 173 180. [7] BUGEAUD,Y & LAURENT,M., Minoration effective e la istance -aique entre uissances e nombres algébriques, J. Number Theory 61, 1996) 311-342. [8] CARMICHAEL,R.D., On the numerical factors of the arithmetic forms α n ± β n, Ann. Math. 15 1913), 30 70. [9] COHN,H., A classical invitation to algebraic numbers an class fiels, Sringer-Verlag, New York, 1978. [10], Introuction to the construction of class fiels, Cambrige stuies in avance mathematics 6, Cambrige University Press, Cambrige, 1985. [11] DICKSON,L.E., History of the theory of numbers, Vol. I, The Carnegie Institute of Washington, New York, 1952. [12] DOBROWOLSKI,E., On a question of Lehmer an the number of irreucible factors of a olynomial, Acta Arith. 34 1979), 391 401. [13] DURST,L.K., Excetional real Lehmer sequences, Pacific J. Math. 9 1959), 437 441. [14] ERDŐS,P., Some recent avances an current roblems in number theory, Lectures on moern mathematics, Vol. III e. T.L. Saaty), Wiley, New York, 1965, 196 244. [15] ERDŐS,P. & SHOREY,T.N., On the greatest rime factor of 2 1 for a rime an other exressions, Acta Arith. 30 1976), 257 265. [16] FRÖHLICH,A. & TAYLOR,M.J., Algebraic number theory, Cambrige stuies in avance mathematics 27, Cambrige University Press, Cambrige, 1991. [17] GELFOND,A.O., Transcenental an algebraic numbers, Dover Publications, New York, 1960. [18] HALBERSTAM,H. & RICHERT,H.E., Sieve methos, Acaemic Press, Lonon, 1974. [19] HARDY,G.H. & WRIGHT,E.M., An introuction to the theory of numbers, Oxfor University Press, 5th e, Oxfor 1979). [20] HECKE,E., Lectures on the theory of algebraic numbers, Grauate Texts in Mathematics 77, Sringer-Verlag, New York, 1981. [21] JURICEVIC,R., Lehmer numbers with at least 2 rimitive ivisors, Ph.D. thesis, University of Waterloo, 2007. [22] LEHMER,D.H., An extene theory of Lucas functions, Ann. Math. 31 1930), 419 448.

18 C.L. STEWART [23] LAGARIAS,J.C. & ODLYZKO,A.M., Effective versions of the Chebotarev Density Theorem, in Algebraic Number Fiels L-functions an Galois roerties), eite by A. Fröhlich, Acaemic Press, Lonon, 1977, 409 464. [24] LUCAS,E., Sur les raorts qui existent entre la théorie es nombres et le calcul intégral, C.R. Aca. Sci. Paris 82 1876), 1303 1305. [25], Théorie es fonctions numériques simlement érioiques, Amer. J. Math. 1 1878), 184 240, 289 321. [26] MATIJASEVIC,Y., Enumerable sets are iohantine Russian), Dokl. Aka. Nauk SSSR 191 1970), 279 282. Imrove English translation: Soviet Math. Doklay 11 1970), 354 357. [27] MURATA,L. & POMERANCE,C., On the largest rime factor of a Mersenne number, CRM Proc. Lecture Notes 36, H. Kisilevsky an E.Z. Goren, eitors, Amer. Math. Soc., Provience, R.I., 2004, 209 218. [28] MURTY,R. & WONG,S., The ABC conjecture an rime ivisors of the Lucas an Lehmer sequences, Number Theory for the Millennium III, eite by M.A. Bennett, B.C. Bernt, N. Boston, H.G. Diamon, A.J. Hilebran an W. Phili, A.K. Peters, Natick, MA, 2002, 43 54. [29] ROTKIEWICZ,A., On Lucas numbers with two intrinsic rime ivisors, Bull. Aca. Polon. Sci. Sér. Math. Astr. Phys. 10 1962), 229 232. [30] SCHINZEL,A., On rimitive rime factors of a n b n, Proc. Cambrige Philos. Soc. 58 1962), 555 562. [31], The intrinsic ivisors of Lehmer numbers in the case of negative iscriminant, Ark. Mat. 4 1962), 413 416. [32], On rimitive rime factors of Lehmer numbers, I, Acta Arith. 8 1963), 213 223. [33], On rimitive rime factors of Lehmer numbers, II, Acta Arith. 8 1963), 251 257. [34], On rimitive rime factors of Lehmer numbers, III, Acta Arith. 15 1968), 49 69. [35], Primitive ivisors of the exression A n B n in algebraic number fiels, J. reine angew. Math. 268/269 1974), 27 33. [36] SHOREY,T.N. & STEWART,C.L., On ivisors of Fermat, Fibonacci, Lucas an Lehmer numbers, II, J. Lonon Math. Soc. 23 1981), 17 23. [37] STEWART,C.L., The greatest rime factor of a n b n, Acta Arith. 26 1975), 427 433. [38], On ivisors of Fermat, Fibonacci, Lucas an Lehmer numbers, Proc. Lonon Math. Soc. 35 1977), 425 447. [39], Primitive ivisors of Lucas an Lehmer numbers, Transcenence theory: avances an alications es. A. Baker an D.W. Masser), Acaemic Press, Lonon, 1977, 79 92. [40], On ivisors of Fermat, Fibonacci, Lucas an Lehmer numbers, III, J. Lonon Math. Soc. 28 1983), 211 217. [41] STEWART,C.L. & YU,K., On the abc conjecture, II, Duke Math. J. 108 2001), 169 181. [42] VOUTIER,P.M., Primitive ivisors of Lucas an Lehmer sequences, II, J. Th. Nombres Boreaux 8 1996), 251 274. [43], Primitive ivisors of Lucas an Lehmer sequences, III, Math. Proc. Camb. Phil. Soc. 123 1998), 407 419. [44], An effective lower boun for the height of algebraic numbers, Acta Arith. 74 1996), 81 95. [45] WARD,M., The intrinsic ivisors of Lehmer numbers, Ann. Math. 62 1955), 230 236. [46] YAMADA,Y., A note on the aer by Bugeau an Laurent Minoration effective e la istance -aique entre uissances e nombres algébriques, J. Number Theory 130, 2010) 1889-1897. [47] YU,K., P aic logarithmic forms an grou varieties III, Forum Math. 19 2007), 187 280. [48], P-aic logarithmic forms an a roblem of Erős, to aear. [49] ZSIGMONDY,K., Zur Theorie er Potenzreste, Monatsh. Math. 3 1892), 265 284. C.L. Stewart, Deartment of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canaa, email: cstewart@uwaterloo.ca