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Sums of S-units in recurrence sequences
Accepted Manuscript Sums of S-units in recurrence sequences
A. Bérczes, L. Hajdu, I. Pink, S.S. Rout
PII: DOI: Reference:
S0022-314X(18)30309-3 https://doi.org/10.1016/j.jnt.2018.10.009 YJNTH 6151
To appear in:
Journal of Number Theory
Received date: Revised date: Accepted date:
20 July 2018 2 October 2018 2 October 2018
Please cite this article in press as: A. Bérczes et al., Sums of S-units in recurrence sequences, J. Number Theory (2019), https://doi.org/10.1016/j.jnt.2018.10.009
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
SUMS OF S-UNITS IN RECURRENCE SEQUENCES ´ A. BERCZES, L. HAJDU, I. PINK, AND S. S. ROUT Abstract. In this paper we give various finiteness results concerning terms of recurrence sequences Un representable as a sum of S-units with a fixed number of terms. We prove that under certain (necessary) conditions, the number of indices n for which Un allows such a representation is finite, and can be bounded in terms of the parameters involved. In this generality, our result is ineffective, i.e. we cannot bound the size of the exceptional indices. We also give an effective result, under some stronger assumptions.
1. Introduction Let S = {p1 , . . . , ps } be a set of primes, and write ZS for the set of those integers which have no prime divisors outside S. Further, let Un be a linear recurrence sequence of integers. In this paper we are interested in the equation (1)
Un = w1 + · · · + wk
with some arbitrary but fixed k ≥ 1, in unknown n ≥ 0 and w1 , . . . , wk ∈ ZS . Our aim is to establish finiteness results for the solutions of (1). There are many results in the literature concerning equation (1). In case of k = 1, a general finiteness result immediately follows from deep theorems of Peth˝o [10] and Shorey and Stewart [15] (obtained independently), concerning perfect powers in recurrence sequences. Peth˝o and de Weger [12] and de Weger [18] provided algorithms and numerical results for the complete solution of (1) in this case, for fixed sequences Un and sets of primes S; see also section III of Mignotte and Tzanakis [9]. In the general case k ≥ 2 much less is known; in particular, there is no general finiteness result concerning this situation. Dealing with binary recurrence sequences, Guzman-Sanchez and Luca [6] noted that under certain technical conditions (namely conditions imposed on the 2010 Mathematics Subject Classification. 11D61. Key words and phrases. Recurrence sequences, sums of S-units, S-unit equations, polynomial-exponential equations, Baker’s method. Research supported in part by the NKFIH grant K115479 and the projects EFOP-3.6.1-16-2016-00022 and EFOP-3.6.2-16-2017-00015 co-financed by the European Union and the European Social Fund. 1
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´ A. BERCZES, L. HAJDU, I. PINK, AND S. S. ROUT
solutions), results obtained by Evertse [3] and independently by van der Poorten and Schlickewei [13], make it possible to obtain finiteness results for n, w1 , . . . , wk in (1). Beside that, some special cases are handled, too. Luca and Szalay [7] gave effective finiteness results for the solutions of (1) when Un is the Fibonacci sequence, and the right hand side is of the form pa ± pb + 1 with p prime. For related, ineffective results see Peth˝o and Tichy [11]. We also mention a recent result of Bert´ok, Hajdu, Pink and R´abai [1] yielding an explicit upper bound under some conditions for the solutions of (1), in the special case where w1 , . . . , wk are prime powers. In case of special sequences complete lists of the solutions are also given: in [1] the cases where Un is one of the Fibonacci sequence, the sequence of Lucas numbers, the Pell sequence and the associate Pell sequence and the right hand side is of the form 2a + 3b + 5c is handled. (See [1] for further related references.) Finally, we mention that there are many results in the literature where instead of sums of powers yielding terms of recurrence sequences just the ”opposite” problem is handled: sums of terms of a recurrence sequence yielding perfect powers. In this direction we only refer to the recent paper of Bravo, Faye and Luca [2], and the references therein. For further related problems concerning S-units and recurrence sequences, see e.g. [5] and [4]. In this paper we provide two finiteness results concerning equation (1), of different types. First we give a general ineffective finiteness result for the number of indices n for which (1) has a solution in w1 , . . . , wk . The conditions we impose are natural and necessary (they concern the recurrence sequence involved, not the solutions), and our theorem (in contrast to the above mentioned remark in [6]) is valid for recurrence sequences of arbitrary order. Then, under certain technical assumptions we provide an effective upper bound for the solutions of (1). This theorem extends the corresponding statement from [1] to this more general situation. 2. Notation and main results A linear recurrence sequence of order r is a sequence (Un )n≥0 satisfying a relation (2)
Un = a1 Un−1 + · · · + ar Un−r
where a1 , . . . , ar are integers with ar = 0 and U0 , . . . , Ur−1 are integers not all zero. For later use, put T := max max |ai |, max |Uj | . 1≤i≤r
0≤j≤r−1
SUMS OF S-UNITS IN RECURRENCE SEQUENCES
3
The characteristic polynomial of Un is defined by t (x − αi )mi (3) f (x) := xr − a1 xr−1 − · · · − ar = i=1
where α1 , . . . , αt are distinct algebraic numbers and m1 , . . . , mt are positive integers. Then as it is well-known (see e.g. Theorem C1 in part C of [17]) we have a representation of the form (4)
Un =
t
gi (n)αin
for all n ≥ 0.
i=1
Here gi (x) is a polynomial of degree mi − 1 (i = 1, . . . , t) with coefficients in the number field Q(α1 , . . . , αt ). The sequence (Un )n≥0 is called degenerate if there are integers i, j with 1 ≤ i < j ≤ t such that αi /αj is a root of unity; otherwise it is called non-degenerate. If for some i with 1 ≤ i ≤ t we have |αi | > |αj | for all j with 1 ≤ j ≤ t and j = i, then we say that αi is a dominant root of the sequence (Un )n≥0 . Now we give our new results. We start with a general (ineffective) finiteness theorem. Theorem 2.1. Let Un be a non-degenerate recurrence sequence of order r, and write f (x) for the characteristic polynomial of Un . Suppose that at least one of the following two conditions is valid: (i) f (x) has at least one irrational root, (ii) f (0) has a prime divisor outside S. Then for any fixed k ≥ 1, equation (1) is solvable at most for finitely many n. Further, the number of indices n for which (1) is solvable for this fixed k, can be bounded by an effectively computable constant depending only on r, s and k. Remark 1. Choosing S = {2, 3}, the binary recurrence sequence Un = 3n − 2n shows that the conditions of Theorem 2.1 are necessary. Indeed, as f (x) = x2 − 5x + 6, neither of the conditions (i) and (ii) is satisfied, and obviously, equation (1) has solutions for all n, already with k = 2. Our next theorem gives, under certain conditions, an effective finiteness result on the solutions of equation (1). Theorem 2.2. Let (Un )n≥0 be a non-degenerate recurrence sequence of order r ≥ 2. Suppose that (Un )n≥0 has a dominant root which is not an integer. Let ε > 0 be arbitrary. Then for all k ≥ 1, for all solutions n, w1 , . . . , wk of (1) satisfying |wi |1+ε < |wk | (i = 1, . . . , k − 1) we have max(n, |w1 |, . . . , |wk |) ≤ C,
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where C is an effectively computable constant depending only on ε, T , r, s, p1 , . . . , ps , k. Remark 2. By choosing Un = 2n − 1 and S = {2}, we see that the equation Un = −1 + 2z has infinitely many solutions given by n = z. This shows that the assumption that the dominant root of Un is not an integer is necessary. 3. Proofs To give the proof of Theorem 2.1, we need two lemmas. The first one concerns polynomial-exponential equations, that is equations of the form
(5)
Pi (x)αxi = 0
i=1
in x = (x1 , . . . , xm ) ∈ Zm , where the Pi are polynomials of total degree bounded by D over a number field K of degree d, and x1 xm . . . αim αxi = αi1
with αij ∈ K \ {0} (1 ≤ i ≤ , 1 ≤ j ≤ m). Set L = {1, . . . , }, and for some fixed partition P of L consider the system of equations Pi (x)αxi = 0 (λ ∈ P). i∈λ
Denote by H(P) the set of those solutions of the above system of equations, which do not satisfy any system of the form Pi (x)αxi = 0 (λ ∈ Q), i∈λ
where Q is a proper refinement of P. Further, let G(P) be the subgroup of Zm consisting of those x for which αxi = αxj whenever i and j lie in the same λ ∈ P. The following result is due to Schlickewei and Schmidt [14]. Lemma 3.1. For any partition P of L for which G(P) = {0} we have |H(P)| < C(m, d, D), where C(m, d, D) is an effectively computable constant depending only on m, d, D. To formulate our second lemma we need to introduce some notation. For an algebraic number η of degree deg(η) over Q, we define as usual
SUMS OF S-UNITS IN RECURRENCE SEQUENCES
5
the absolute logarithmic height of η by the formula ⎞ ⎛ deg(η) 1 ⎝ log max 1, |η (j) | ⎠ , log |a0 | + (6) h(η) = deg(η) j=1 where a0 is the leading coefficient of the minimal polynomial of η over Z and the η (j) -s are the conjugates of η. We recall the following wellknown properties of the absolute logarithmic height: h(η1 + · · · + ηm ) ≤ h(η1 ) + · · · + h(ηm ) + log m, h(η1 · · · ηm ) ≤ h(η1 ) + · · · + h(ηm ), h(η ) = ||h(η) ( ∈ Z). The next lemma gives important information about the parameters appearing in the explicit forms of recurrence sequences. Lemma 3.2. Let (Un )n≥0 be a non-degenerate recurrence sequence of order r ≥ 2. (i) With the notation in (4), write gi (n) = βi,0 + βi,1 n + · · · + βi,mi −1 nmi −1 (i = 1, . . . , t). Then we have max
1≤i≤t, 0≤≤mi −1
{h(αi ), h(βi, )} ≤ c1 .
Here c1 is an effectively computable constant depending only on T and r. (ii) If n ≥ 1 and gi (n) = 0 then c2 ≤ |gi (n)| ≤ c3 nmi −1
(1 ≤ i ≤ t),
where c2 and c3 are effectively computable constants depending only on T and r. (iii) Suppose that α1 is dominant root of the sequence (Un )n≥0 . Then we have |Un | ≤ c4 nr−1 |α1 |n (n ≥ 1), where c4 is an effectively computable constant depending only on T and r. (iv) Suppose that (Un )n≥0 has a dominant root. Let n, w1 , . . . , wk be a solution of equation (1) such that (7)
|wi |1+ε ≤ |wk |
(1 ≤ i ≤ k − 1)
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´ A. BERCZES, L. HAJDU, I. PINK, AND S. S. ROUT
with some ε > 0. Then log |wk | < c5 n + c6 , where c5 and c6 are effectively computable constants depending only on ε, T , r and k. Proof. Part (i) is well-known; see e.g. the arguments given in part C of [17]. Part (ii) is also long known; it is a trivial consequence of (i). Part (iii) is obvious. We turn to the proof of (iv). By (1) and (7) we get 1
ε
|wk | 1+ε (|wk | 1+ε − (k − 1)) ≤ |Un |. ε
We may clearly assume that |wk | 1+ε > k. Thus 1
|wk | 1+ε < |Un |. As the sequence (Un )n≥0 has a dominant root, the statement follows from (iii). Proof of Theorem 2.1. Consider equation (1) with some k ≥ 1. Using (4), this equation can be rewritten as (8)
t
gi (n)αin = pz111 · · · pzs1s + · · · + pz1k1 · · · pzsks
i=1
in non-negative integers n, zuv (1 ≤ u ≤ k, 1 ≤ v ≤ s). Let P be any partition of the set {1, . . . , t + k}. Suppose that (i) is valid. Consider first those solutions of (8) which do not satisfy any refinement of P, and for which there exists i, j with 1 ≤ i < j ≤ t such that the terms gi (n)αin and gj (n)αjn belong to the same λ ∈ P. Since Un is non-degenerate, we conclude that G(P) = {0}. With Lemma 3.1, the boundedness of the number of such solutions is immediate. Consider now those solutions of (8) which still do not satisfy any refinement of P, and for which the terms gi (n)αin (1 ≤ i ≤ t) belong to different classes of P. Take a class λ ∈ P such that gi (n)αin belongs to λ with some irrational αi . There exists an isomorphism σ of Q(α1 , . . . , αt ) into C such that σ(αi ) = αj with some j = i. As gi (n)αin = w1 +· · ·+wh for some {1 , . . . , h } ⊂ {1, . . . , k}, this implies gi (n)αin = σ(gi (n))αjn . However, by Lemma 3.1 this equation has only finitely many solutions, and their number can be effectively bounded. Since the number of partitions P of {1, . . . , t + k} is bounded in terms of t and k, the theorem is valid in case (i). Assume now that (ii) holds. In view of what we have proved so far, we may suppose that all the roots αi of f (x) are rational integers. Consider those solutions of (1) which do not satisfy any refinement of P. Similarly as in case of (i) we see that all classes P contain at most
SUMS OF S-UNITS IN RECURRENCE SEQUENCES
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one term of the form gi (n)αin . Now (ii) implies that there is an index i such that p | αi for some prime p outside S. This immediately shows that G(P) = {0}. Hence Lemma 3.1 implies that the number of such solutions is bounded. This as before, shows that the statement is valid also in this case, and the theorem follows. To prove Theorem 2.2 we also need the next lemma, which is a Baker-type result of Matveev [8]. Lemma 3.3 (Matveev [8]). Denote by η1 , . . . , ηm algebraic numbers, not 0 or 1, by log η1 , . . . , log ηm the principal values of their logarithms, by D the degree of the number field K = Q(η1 , . . . , ηm ) over Q, and by b1 , . . . , bm rational integers. Define B = max{|b1 |, . . . , |bm |}, and Ai = max{Dh(ηi ), | log ηi |, 0.16}
(1 ≤ i ≤ m),
where h(η) denotes the absolute logarithmic height of η. Consider the linear form Λ = b1 log η1 + · · · + bm log ηm and assume that Λ = 0. Then log |Λ| ≥ −C(m, κ)D2 A1 · · · Am log(eD) log(eB), where κ = 1 if K ⊂ R and κ = 2 otherwise and
κ 1 1 m+3 3.5 6m+20 em 30 . m ,2 C(m, κ) = min κ 2 Proof of Theorem 2.2. Consider the solutions of equation (1) satisfying the assumptions of the theorem. We shall give an effective upper bound for max(n, |w1 |, . . . , |wk |). In the proof c7 , . . . , c17 denote positive effective constants depending only on ε, T, r, s, p1 , . . . , ps , k. Note that once we have an effective upper bound for n then (iv) of Lemma 3.2 immediately implies an upper bound for max(|w1 |, . . . , |wk |), as well. Therefore it is enough to derive an effective upper bound only for n. In particular, in the rest of the proof we shall assume that n > 0, since in case of n = 0 we are immediately done. Let α1 be the dominant root of (Un )n≥0 . If either gi (n) = 0 for some 1 ≤ i ≤ t or |Un | ≤ 12 |g1 (n)||α1 |n we can obtain easily an effective upper bound for n. Indeed, if gi (n) = 0 holds for some 1 ≤ i ≤ t then (i) of Lemma 3.2 gives that n ≤ c7 . Further, if |Un | ≤ 12 |g1 (n)||α1 |n (and
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´ A. BERCZES, L. HAJDU, I. PINK, AND S. S. ROUT
gi (n) = 0, (1 ≤ i ≤ t) ) we get by (4) that t t 1 n n ≤ |g1 (n)||α1 | ≤ gi (n)αi |gi (n)||αin |, 2 i=2
i=2
which in view of the fact that α1 is the dominant root, by (ii) of Lemma 3.2 leads to n ≤ c8 . Therefore, in what follows, we may assume that 1 (9) gi (n) = 0, (1 ≤ i ≤ t) and |Un | > |g1 (n)||α1 |n . 2 Now, using (4) we can rewrite (1) as t k−1 n j=2 gj (n)αj n −1 i=1 wi − , g1 (n)α1 wk − 1 = wk wk whence putting Φ := g1 (n)α1n wk−1 , we obtain t k−1 n j=2 |gj (n)||αj | i=1 |wi | + . (10) |Φ − 1| ≤ |wk | |wk | 1
Since, by assumption, for every 1 ≤ i ≤ k − 1 we have |wi | ≤ |wk | 1+ε , we get by (1) and (9) that 1 (11) |wk | > |g1 (n)||α1 |n . 2k The combination of (9) and (11) yields k−1 ε k−1 (k − 1)(2k) 1+ε i=1 |wi | ≤ < . (12) ε ε ε |wk | |wk | 1+ε |g1 (n)| 1+ε |α1 | 1+ε n Hence, using (12) and (ii) of Lemma 3.2 we obtain that k−1 c9 i=1 |wi | < (13) . ε |wk | |α1 | 1+ε n t
|gj (n)||αj |n
Now, we derive an upper bound for j=2 |wk | . Without loss of generality we may assume that |α2 | ≥ · · · ≥ |αt |. Using (11) and (ii) of Lemma 3.2 we may write that t n n tc3 nr−1 |α2 |n j=2 |gj (n)||αj | r−1 α2 < 1 < c10 n . (14) n |wk | α1 |g (n)||α1 | 2k 1 Thus (10), (13) and (14) imply r−1 max (15) |Φ − 1| < c11 n
n α2 1 . ε , |α1 | 1+ε α1
SUMS OF S-UNITS IN RECURRENCE SEQUENCES
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In what follows, we distinguish two cases according to |Φ − 1| > 12 or 0 ≤ |Φ − 1| ≤ 12 , respectively. If |Φ − 1| > 12 then the combination of (15) with the fact that α1 is the dominant root leads easily to an upper bound n ≤ c12 . So, we may assume that 0 ≤ |Φ − 1| ≤ 12 . If now |Φ − 1| = 0, that is Φ = 1, we get by the definition of Φ that g1 (n)α1n = wk .
(16)
Since α1 ∈ Z then there exists a conjugate αi of α1 in the field K = Q(α1 , . . . , αt ) such that α1 = αi . Therefore, on taking the i-th conjugate of both sides of (16) and observing that wk is a rational integer, we may write that gi (n)αin = wk . Thus n α1 gi (n) = αi g1 (n) , which by (ii) of Lemma 3.2 yields that n ≤ c13 . Finally, we may assume that 0 < |Φ − 1| ≤ 12 . It is well known (see e. g. Lemma B.2 of Appendix B of [16]) that if |Φ − 1| ≤ 12 we have | log(Φ)| < 2|Φ − 1|, where log(Φ) denotes the principal value of the logarithm of the complex number Φ. Hence by (15) we obtain that n α2 1 r−1 (17) |Λ| < 2|Φ − 1| = 2c11 n max ε , |α | 1+ε α1 1
where Λ := log(Φ). Since wi ∈ ZS (1 ≤ i ≤ k), we may write |wk | in the form |wk | = pz1k1 . . . pzsks , where the numbers zkj are non-negative integers. Further, since Φ = g1 (n)α1n wk−1 we get that Λ=
s
(−zki ) log(pi ) + log(g1 (n)) + n log(α1 ) + b0 log(−1),
i=1
where b0 is an integer with |b0 | ≤ s + 2. In order to give a non-trivial lower bound for |Λ| we shall use Lemma 3.3. Since Φ = 1 we also have that Λ = 0. Hence we can apply Lemma 3.3 to Λ with K = Q(α1 , . . . , αt ), D ≤ rt , m = s + 3, ηj = pj (1 ≤ j ≤ s), ηs+1 = g1 (n), ηs+2 = α1 , ηs+3 = −1. By (iv) of Lemma 3.2 we easily see that we may choose B = c14 n with an effective constant c14 . Further, it is clear that for 1 ≤ j ≤ s the choice Aj = log pj is suitable, moreover by (ii) of Lemma 3.2 there exists an effective constant c15 such that max{As+1 , As+2 , As+3 } ≤ c15 log n. By applying Lemma 3.3 with the above parameters we obtain that (18)
|Λ| > exp(−c16 log n).
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´ A. BERCZES, L. HAJDU, I. PINK, AND S. S. ROUT
On comparing (18) with (17), we get that n ≤ c17 . Thus our theorem is proved. 4. Acknowledgements The authors are grateful to the referee for her/his useful and helpful comments. This work was started when the last author visited Institute of Mathematics, University of Debrecen and he thanks the people of this institute for their hospitality and support. He also thanks HarishChandra Research Institute Allahabad to financially support for this research visit.
References [1] Cs. Bert´ok, L. Hajdu, I. Pink and Zs. R´ abai, Linear combinations of prime powers in binary recurrence sequences, Int. J. Number Theory 13 (2017), 261– 271. [2] J. J. Bravo, B. Faye and F. Luca, Powers of two as sums of three Pell numbers, Taiwanese J. Math. 21 (2017), 739–751. [3] J.-H. Evertse, On sums of S-units and linear recurrences, Compositio Math. 53 (1984), 225–244. [4] J.-H. Evertse and K. Gy˝ory, Unit Equations in Diophantine Number Theory, Cambridge University Press, 2015, pp. 378. [5] J.-H. Evertse, K. Gy˝ory, C. Stewart and R. Tijdeman, S-unit equations and their applications, New Advances in Transcendence Theory (A. Baker, ed.), Cambridge University Press, Cambridge, 1988, 110–174. [6] S. Guzman-Sanchez and F. Luca, Linear combinations of factorials and S-units in a binary recurrence sequence, Ann. Math. Qu´e. 38 (2014), 169–188. [7] F. Luca and L. Szalay, Fibonacci numbers of the form pa ± pb + 1, Fibonacci Quart. 45 (2007), 98–103. [8] E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers, Izv. Math. 64 (2000), 1217–1269. [9] M. Mignotte and N. Tzanakis, Arithmetical study of recurrence sequences, Acta Arith. 57 (1991), 357–364. [10] A. Peth˝o, Perfect powers in second order linear recurrences, J. Number Theory 15 (1982) 5–13. [11] A. Peth˝o and R. F. Tichy, S-unit equations, linear recurrences and digit expansions, Publ. Math. Debrecen 42 (1993) 145–154. [12] A. Peth˝o and B. M. M. de Weger, Products of prime powers in binary recurrence sequences I. The hyperbolic case, with an application to the generalized Ramanujan-Nagell equation, Math. Comp. 47/176 (1986), 713–727. [13] A. J. van der Poorten and H. P. Schlickewei, The growth conditions for recurrence sequences, Macquarie Univ. Math. Rep. 82–0041. North Ryde, Australia (1982). [14] H. P. Schlickewei and W. M. Schmidt, The number of solutions of polynomialexponential equations, Compositio Math. 120 (2000), 193–225.
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[15] T. N. Shorey and C. L. Stewart, On the Diophantine equation ax2t + bxt y + cy 2 = d and pure powers in recurrence sequences, Math. Scand. 52 (1983), 24–36. [16] N. P. Smart, The algorithmic resolution of Diophantine equations, London Mathematical Society Student Texts Vol. 41, Cambridge University Press, Cambridge, 1998, xvi+243. [17] T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, Cambridge, 1986. [18] B. M. M. de Weger Products of prime powers in binary recurrence sequences II. The elliptic case, with an application to a mixed quadratic-exponential equation, Math. Comp. 47 (1986), 729–739.
´rczes, L. Hajdu, I. Pink A. Be University of Debrecen, Institute of Mathematics H-4002 Debrecen, P.O. Box 400. Hungary S. S. Rout Institute of Mathematics & Applications, Andharua, Bhubaneswar-751029, Odisha, India E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]
A. Bérczes, L. Hajdu, I. Pink, S.S. Rout
PII: DOI: Reference:
S0022-314X(18)30309-3 https://doi.org/10.1016/j.jnt.2018.10.009 YJNTH 6151
To appear in:
Journal of Number Theory
Received date: Revised date: Accepted date:
20 July 2018 2 October 2018 2 October 2018
Please cite this article in press as: A. Bérczes et al., Sums of S-units in recurrence sequences, J. Number Theory (2019), https://doi.org/10.1016/j.jnt.2018.10.009
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
SUMS OF S-UNITS IN RECURRENCE SEQUENCES ´ A. BERCZES, L. HAJDU, I. PINK, AND S. S. ROUT Abstract. In this paper we give various finiteness results concerning terms of recurrence sequences Un representable as a sum of S-units with a fixed number of terms. We prove that under certain (necessary) conditions, the number of indices n for which Un allows such a representation is finite, and can be bounded in terms of the parameters involved. In this generality, our result is ineffective, i.e. we cannot bound the size of the exceptional indices. We also give an effective result, under some stronger assumptions.
1. Introduction Let S = {p1 , . . . , ps } be a set of primes, and write ZS for the set of those integers which have no prime divisors outside S. Further, let Un be a linear recurrence sequence of integers. In this paper we are interested in the equation (1)
Un = w1 + · · · + wk
with some arbitrary but fixed k ≥ 1, in unknown n ≥ 0 and w1 , . . . , wk ∈ ZS . Our aim is to establish finiteness results for the solutions of (1). There are many results in the literature concerning equation (1). In case of k = 1, a general finiteness result immediately follows from deep theorems of Peth˝o [10] and Shorey and Stewart [15] (obtained independently), concerning perfect powers in recurrence sequences. Peth˝o and de Weger [12] and de Weger [18] provided algorithms and numerical results for the complete solution of (1) in this case, for fixed sequences Un and sets of primes S; see also section III of Mignotte and Tzanakis [9]. In the general case k ≥ 2 much less is known; in particular, there is no general finiteness result concerning this situation. Dealing with binary recurrence sequences, Guzman-Sanchez and Luca [6] noted that under certain technical conditions (namely conditions imposed on the 2010 Mathematics Subject Classification. 11D61. Key words and phrases. Recurrence sequences, sums of S-units, S-unit equations, polynomial-exponential equations, Baker’s method. Research supported in part by the NKFIH grant K115479 and the projects EFOP-3.6.1-16-2016-00022 and EFOP-3.6.2-16-2017-00015 co-financed by the European Union and the European Social Fund. 1
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´ A. BERCZES, L. HAJDU, I. PINK, AND S. S. ROUT
solutions), results obtained by Evertse [3] and independently by van der Poorten and Schlickewei [13], make it possible to obtain finiteness results for n, w1 , . . . , wk in (1). Beside that, some special cases are handled, too. Luca and Szalay [7] gave effective finiteness results for the solutions of (1) when Un is the Fibonacci sequence, and the right hand side is of the form pa ± pb + 1 with p prime. For related, ineffective results see Peth˝o and Tichy [11]. We also mention a recent result of Bert´ok, Hajdu, Pink and R´abai [1] yielding an explicit upper bound under some conditions for the solutions of (1), in the special case where w1 , . . . , wk are prime powers. In case of special sequences complete lists of the solutions are also given: in [1] the cases where Un is one of the Fibonacci sequence, the sequence of Lucas numbers, the Pell sequence and the associate Pell sequence and the right hand side is of the form 2a + 3b + 5c is handled. (See [1] for further related references.) Finally, we mention that there are many results in the literature where instead of sums of powers yielding terms of recurrence sequences just the ”opposite” problem is handled: sums of terms of a recurrence sequence yielding perfect powers. In this direction we only refer to the recent paper of Bravo, Faye and Luca [2], and the references therein. For further related problems concerning S-units and recurrence sequences, see e.g. [5] and [4]. In this paper we provide two finiteness results concerning equation (1), of different types. First we give a general ineffective finiteness result for the number of indices n for which (1) has a solution in w1 , . . . , wk . The conditions we impose are natural and necessary (they concern the recurrence sequence involved, not the solutions), and our theorem (in contrast to the above mentioned remark in [6]) is valid for recurrence sequences of arbitrary order. Then, under certain technical assumptions we provide an effective upper bound for the solutions of (1). This theorem extends the corresponding statement from [1] to this more general situation. 2. Notation and main results A linear recurrence sequence of order r is a sequence (Un )n≥0 satisfying a relation (2)
Un = a1 Un−1 + · · · + ar Un−r
where a1 , . . . , ar are integers with ar = 0 and U0 , . . . , Ur−1 are integers not all zero. For later use, put T := max max |ai |, max |Uj | . 1≤i≤r
0≤j≤r−1
SUMS OF S-UNITS IN RECURRENCE SEQUENCES
3
The characteristic polynomial of Un is defined by t (x − αi )mi (3) f (x) := xr − a1 xr−1 − · · · − ar = i=1
where α1 , . . . , αt are distinct algebraic numbers and m1 , . . . , mt are positive integers. Then as it is well-known (see e.g. Theorem C1 in part C of [17]) we have a representation of the form (4)
Un =
t
gi (n)αin
for all n ≥ 0.
i=1
Here gi (x) is a polynomial of degree mi − 1 (i = 1, . . . , t) with coefficients in the number field Q(α1 , . . . , αt ). The sequence (Un )n≥0 is called degenerate if there are integers i, j with 1 ≤ i < j ≤ t such that αi /αj is a root of unity; otherwise it is called non-degenerate. If for some i with 1 ≤ i ≤ t we have |αi | > |αj | for all j with 1 ≤ j ≤ t and j = i, then we say that αi is a dominant root of the sequence (Un )n≥0 . Now we give our new results. We start with a general (ineffective) finiteness theorem. Theorem 2.1. Let Un be a non-degenerate recurrence sequence of order r, and write f (x) for the characteristic polynomial of Un . Suppose that at least one of the following two conditions is valid: (i) f (x) has at least one irrational root, (ii) f (0) has a prime divisor outside S. Then for any fixed k ≥ 1, equation (1) is solvable at most for finitely many n. Further, the number of indices n for which (1) is solvable for this fixed k, can be bounded by an effectively computable constant depending only on r, s and k. Remark 1. Choosing S = {2, 3}, the binary recurrence sequence Un = 3n − 2n shows that the conditions of Theorem 2.1 are necessary. Indeed, as f (x) = x2 − 5x + 6, neither of the conditions (i) and (ii) is satisfied, and obviously, equation (1) has solutions for all n, already with k = 2. Our next theorem gives, under certain conditions, an effective finiteness result on the solutions of equation (1). Theorem 2.2. Let (Un )n≥0 be a non-degenerate recurrence sequence of order r ≥ 2. Suppose that (Un )n≥0 has a dominant root which is not an integer. Let ε > 0 be arbitrary. Then for all k ≥ 1, for all solutions n, w1 , . . . , wk of (1) satisfying |wi |1+ε < |wk | (i = 1, . . . , k − 1) we have max(n, |w1 |, . . . , |wk |) ≤ C,
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´ A. BERCZES, L. HAJDU, I. PINK, AND S. S. ROUT
where C is an effectively computable constant depending only on ε, T , r, s, p1 , . . . , ps , k. Remark 2. By choosing Un = 2n − 1 and S = {2}, we see that the equation Un = −1 + 2z has infinitely many solutions given by n = z. This shows that the assumption that the dominant root of Un is not an integer is necessary. 3. Proofs To give the proof of Theorem 2.1, we need two lemmas. The first one concerns polynomial-exponential equations, that is equations of the form
(5)
Pi (x)αxi = 0
i=1
in x = (x1 , . . . , xm ) ∈ Zm , where the Pi are polynomials of total degree bounded by D over a number field K of degree d, and x1 xm . . . αim αxi = αi1
with αij ∈ K \ {0} (1 ≤ i ≤ , 1 ≤ j ≤ m). Set L = {1, . . . , }, and for some fixed partition P of L consider the system of equations Pi (x)αxi = 0 (λ ∈ P). i∈λ
Denote by H(P) the set of those solutions of the above system of equations, which do not satisfy any system of the form Pi (x)αxi = 0 (λ ∈ Q), i∈λ
where Q is a proper refinement of P. Further, let G(P) be the subgroup of Zm consisting of those x for which αxi = αxj whenever i and j lie in the same λ ∈ P. The following result is due to Schlickewei and Schmidt [14]. Lemma 3.1. For any partition P of L for which G(P) = {0} we have |H(P)| < C(m, d, D), where C(m, d, D) is an effectively computable constant depending only on m, d, D. To formulate our second lemma we need to introduce some notation. For an algebraic number η of degree deg(η) over Q, we define as usual
SUMS OF S-UNITS IN RECURRENCE SEQUENCES
5
the absolute logarithmic height of η by the formula ⎞ ⎛ deg(η) 1 ⎝ log max 1, |η (j) | ⎠ , log |a0 | + (6) h(η) = deg(η) j=1 where a0 is the leading coefficient of the minimal polynomial of η over Z and the η (j) -s are the conjugates of η. We recall the following wellknown properties of the absolute logarithmic height: h(η1 + · · · + ηm ) ≤ h(η1 ) + · · · + h(ηm ) + log m, h(η1 · · · ηm ) ≤ h(η1 ) + · · · + h(ηm ), h(η ) = ||h(η) ( ∈ Z). The next lemma gives important information about the parameters appearing in the explicit forms of recurrence sequences. Lemma 3.2. Let (Un )n≥0 be a non-degenerate recurrence sequence of order r ≥ 2. (i) With the notation in (4), write gi (n) = βi,0 + βi,1 n + · · · + βi,mi −1 nmi −1 (i = 1, . . . , t). Then we have max
1≤i≤t, 0≤≤mi −1
{h(αi ), h(βi, )} ≤ c1 .
Here c1 is an effectively computable constant depending only on T and r. (ii) If n ≥ 1 and gi (n) = 0 then c2 ≤ |gi (n)| ≤ c3 nmi −1
(1 ≤ i ≤ t),
where c2 and c3 are effectively computable constants depending only on T and r. (iii) Suppose that α1 is dominant root of the sequence (Un )n≥0 . Then we have |Un | ≤ c4 nr−1 |α1 |n (n ≥ 1), where c4 is an effectively computable constant depending only on T and r. (iv) Suppose that (Un )n≥0 has a dominant root. Let n, w1 , . . . , wk be a solution of equation (1) such that (7)
|wi |1+ε ≤ |wk |
(1 ≤ i ≤ k − 1)
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´ A. BERCZES, L. HAJDU, I. PINK, AND S. S. ROUT
with some ε > 0. Then log |wk | < c5 n + c6 , where c5 and c6 are effectively computable constants depending only on ε, T , r and k. Proof. Part (i) is well-known; see e.g. the arguments given in part C of [17]. Part (ii) is also long known; it is a trivial consequence of (i). Part (iii) is obvious. We turn to the proof of (iv). By (1) and (7) we get 1
ε
|wk | 1+ε (|wk | 1+ε − (k − 1)) ≤ |Un |. ε
We may clearly assume that |wk | 1+ε > k. Thus 1
|wk | 1+ε < |Un |. As the sequence (Un )n≥0 has a dominant root, the statement follows from (iii). Proof of Theorem 2.1. Consider equation (1) with some k ≥ 1. Using (4), this equation can be rewritten as (8)
t
gi (n)αin = pz111 · · · pzs1s + · · · + pz1k1 · · · pzsks
i=1
in non-negative integers n, zuv (1 ≤ u ≤ k, 1 ≤ v ≤ s). Let P be any partition of the set {1, . . . , t + k}. Suppose that (i) is valid. Consider first those solutions of (8) which do not satisfy any refinement of P, and for which there exists i, j with 1 ≤ i < j ≤ t such that the terms gi (n)αin and gj (n)αjn belong to the same λ ∈ P. Since Un is non-degenerate, we conclude that G(P) = {0}. With Lemma 3.1, the boundedness of the number of such solutions is immediate. Consider now those solutions of (8) which still do not satisfy any refinement of P, and for which the terms gi (n)αin (1 ≤ i ≤ t) belong to different classes of P. Take a class λ ∈ P such that gi (n)αin belongs to λ with some irrational αi . There exists an isomorphism σ of Q(α1 , . . . , αt ) into C such that σ(αi ) = αj with some j = i. As gi (n)αin = w1 +· · ·+wh for some {1 , . . . , h } ⊂ {1, . . . , k}, this implies gi (n)αin = σ(gi (n))αjn . However, by Lemma 3.1 this equation has only finitely many solutions, and their number can be effectively bounded. Since the number of partitions P of {1, . . . , t + k} is bounded in terms of t and k, the theorem is valid in case (i). Assume now that (ii) holds. In view of what we have proved so far, we may suppose that all the roots αi of f (x) are rational integers. Consider those solutions of (1) which do not satisfy any refinement of P. Similarly as in case of (i) we see that all classes P contain at most
SUMS OF S-UNITS IN RECURRENCE SEQUENCES
7
one term of the form gi (n)αin . Now (ii) implies that there is an index i such that p | αi for some prime p outside S. This immediately shows that G(P) = {0}. Hence Lemma 3.1 implies that the number of such solutions is bounded. This as before, shows that the statement is valid also in this case, and the theorem follows. To prove Theorem 2.2 we also need the next lemma, which is a Baker-type result of Matveev [8]. Lemma 3.3 (Matveev [8]). Denote by η1 , . . . , ηm algebraic numbers, not 0 or 1, by log η1 , . . . , log ηm the principal values of their logarithms, by D the degree of the number field K = Q(η1 , . . . , ηm ) over Q, and by b1 , . . . , bm rational integers. Define B = max{|b1 |, . . . , |bm |}, and Ai = max{Dh(ηi ), | log ηi |, 0.16}
(1 ≤ i ≤ m),
where h(η) denotes the absolute logarithmic height of η. Consider the linear form Λ = b1 log η1 + · · · + bm log ηm and assume that Λ = 0. Then log |Λ| ≥ −C(m, κ)D2 A1 · · · Am log(eD) log(eB), where κ = 1 if K ⊂ R and κ = 2 otherwise and
κ 1 1 m+3 3.5 6m+20 em 30 . m ,2 C(m, κ) = min κ 2 Proof of Theorem 2.2. Consider the solutions of equation (1) satisfying the assumptions of the theorem. We shall give an effective upper bound for max(n, |w1 |, . . . , |wk |). In the proof c7 , . . . , c17 denote positive effective constants depending only on ε, T, r, s, p1 , . . . , ps , k. Note that once we have an effective upper bound for n then (iv) of Lemma 3.2 immediately implies an upper bound for max(|w1 |, . . . , |wk |), as well. Therefore it is enough to derive an effective upper bound only for n. In particular, in the rest of the proof we shall assume that n > 0, since in case of n = 0 we are immediately done. Let α1 be the dominant root of (Un )n≥0 . If either gi (n) = 0 for some 1 ≤ i ≤ t or |Un | ≤ 12 |g1 (n)||α1 |n we can obtain easily an effective upper bound for n. Indeed, if gi (n) = 0 holds for some 1 ≤ i ≤ t then (i) of Lemma 3.2 gives that n ≤ c7 . Further, if |Un | ≤ 12 |g1 (n)||α1 |n (and
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´ A. BERCZES, L. HAJDU, I. PINK, AND S. S. ROUT
gi (n) = 0, (1 ≤ i ≤ t) ) we get by (4) that t t 1 n n ≤ |g1 (n)||α1 | ≤ gi (n)αi |gi (n)||αin |, 2 i=2
i=2
which in view of the fact that α1 is the dominant root, by (ii) of Lemma 3.2 leads to n ≤ c8 . Therefore, in what follows, we may assume that 1 (9) gi (n) = 0, (1 ≤ i ≤ t) and |Un | > |g1 (n)||α1 |n . 2 Now, using (4) we can rewrite (1) as t k−1 n j=2 gj (n)αj n −1 i=1 wi − , g1 (n)α1 wk − 1 = wk wk whence putting Φ := g1 (n)α1n wk−1 , we obtain t k−1 n j=2 |gj (n)||αj | i=1 |wi | + . (10) |Φ − 1| ≤ |wk | |wk | 1
Since, by assumption, for every 1 ≤ i ≤ k − 1 we have |wi | ≤ |wk | 1+ε , we get by (1) and (9) that 1 (11) |wk | > |g1 (n)||α1 |n . 2k The combination of (9) and (11) yields k−1 ε k−1 (k − 1)(2k) 1+ε i=1 |wi | ≤ < . (12) ε ε ε |wk | |wk | 1+ε |g1 (n)| 1+ε |α1 | 1+ε n Hence, using (12) and (ii) of Lemma 3.2 we obtain that k−1 c9 i=1 |wi | < (13) . ε |wk | |α1 | 1+ε n t
|gj (n)||αj |n
Now, we derive an upper bound for j=2 |wk | . Without loss of generality we may assume that |α2 | ≥ · · · ≥ |αt |. Using (11) and (ii) of Lemma 3.2 we may write that t n n tc3 nr−1 |α2 |n j=2 |gj (n)||αj | r−1 α2 < 1 < c10 n . (14) n |wk | α1 |g (n)||α1 | 2k 1 Thus (10), (13) and (14) imply r−1 max (15) |Φ − 1| < c11 n
n α2 1 . ε , |α1 | 1+ε α1
SUMS OF S-UNITS IN RECURRENCE SEQUENCES
9
In what follows, we distinguish two cases according to |Φ − 1| > 12 or 0 ≤ |Φ − 1| ≤ 12 , respectively. If |Φ − 1| > 12 then the combination of (15) with the fact that α1 is the dominant root leads easily to an upper bound n ≤ c12 . So, we may assume that 0 ≤ |Φ − 1| ≤ 12 . If now |Φ − 1| = 0, that is Φ = 1, we get by the definition of Φ that g1 (n)α1n = wk .
(16)
Since α1 ∈ Z then there exists a conjugate αi of α1 in the field K = Q(α1 , . . . , αt ) such that α1 = αi . Therefore, on taking the i-th conjugate of both sides of (16) and observing that wk is a rational integer, we may write that gi (n)αin = wk . Thus n α1 gi (n) = αi g1 (n) , which by (ii) of Lemma 3.2 yields that n ≤ c13 . Finally, we may assume that 0 < |Φ − 1| ≤ 12 . It is well known (see e. g. Lemma B.2 of Appendix B of [16]) that if |Φ − 1| ≤ 12 we have | log(Φ)| < 2|Φ − 1|, where log(Φ) denotes the principal value of the logarithm of the complex number Φ. Hence by (15) we obtain that n α2 1 r−1 (17) |Λ| < 2|Φ − 1| = 2c11 n max ε , |α | 1+ε α1 1
where Λ := log(Φ). Since wi ∈ ZS (1 ≤ i ≤ k), we may write |wk | in the form |wk | = pz1k1 . . . pzsks , where the numbers zkj are non-negative integers. Further, since Φ = g1 (n)α1n wk−1 we get that Λ=
s
(−zki ) log(pi ) + log(g1 (n)) + n log(α1 ) + b0 log(−1),
i=1
where b0 is an integer with |b0 | ≤ s + 2. In order to give a non-trivial lower bound for |Λ| we shall use Lemma 3.3. Since Φ = 1 we also have that Λ = 0. Hence we can apply Lemma 3.3 to Λ with K = Q(α1 , . . . , αt ), D ≤ rt , m = s + 3, ηj = pj (1 ≤ j ≤ s), ηs+1 = g1 (n), ηs+2 = α1 , ηs+3 = −1. By (iv) of Lemma 3.2 we easily see that we may choose B = c14 n with an effective constant c14 . Further, it is clear that for 1 ≤ j ≤ s the choice Aj = log pj is suitable, moreover by (ii) of Lemma 3.2 there exists an effective constant c15 such that max{As+1 , As+2 , As+3 } ≤ c15 log n. By applying Lemma 3.3 with the above parameters we obtain that (18)
|Λ| > exp(−c16 log n).
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´ A. BERCZES, L. HAJDU, I. PINK, AND S. S. ROUT
On comparing (18) with (17), we get that n ≤ c17 . Thus our theorem is proved. 4. Acknowledgements The authors are grateful to the referee for her/his useful and helpful comments. This work was started when the last author visited Institute of Mathematics, University of Debrecen and he thanks the people of this institute for their hospitality and support. He also thanks HarishChandra Research Institute Allahabad to financially support for this research visit.
References [1] Cs. Bert´ok, L. Hajdu, I. Pink and Zs. R´ abai, Linear combinations of prime powers in binary recurrence sequences, Int. J. Number Theory 13 (2017), 261– 271. [2] J. J. Bravo, B. Faye and F. Luca, Powers of two as sums of three Pell numbers, Taiwanese J. Math. 21 (2017), 739–751. [3] J.-H. Evertse, On sums of S-units and linear recurrences, Compositio Math. 53 (1984), 225–244. [4] J.-H. Evertse and K. Gy˝ory, Unit Equations in Diophantine Number Theory, Cambridge University Press, 2015, pp. 378. [5] J.-H. Evertse, K. Gy˝ory, C. Stewart and R. Tijdeman, S-unit equations and their applications, New Advances in Transcendence Theory (A. Baker, ed.), Cambridge University Press, Cambridge, 1988, 110–174. [6] S. Guzman-Sanchez and F. Luca, Linear combinations of factorials and S-units in a binary recurrence sequence, Ann. Math. Qu´e. 38 (2014), 169–188. [7] F. Luca and L. Szalay, Fibonacci numbers of the form pa ± pb + 1, Fibonacci Quart. 45 (2007), 98–103. [8] E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers, Izv. Math. 64 (2000), 1217–1269. [9] M. Mignotte and N. Tzanakis, Arithmetical study of recurrence sequences, Acta Arith. 57 (1991), 357–364. [10] A. Peth˝o, Perfect powers in second order linear recurrences, J. Number Theory 15 (1982) 5–13. [11] A. Peth˝o and R. F. Tichy, S-unit equations, linear recurrences and digit expansions, Publ. Math. Debrecen 42 (1993) 145–154. [12] A. Peth˝o and B. M. M. de Weger, Products of prime powers in binary recurrence sequences I. The hyperbolic case, with an application to the generalized Ramanujan-Nagell equation, Math. Comp. 47/176 (1986), 713–727. [13] A. J. van der Poorten and H. P. Schlickewei, The growth conditions for recurrence sequences, Macquarie Univ. Math. Rep. 82–0041. North Ryde, Australia (1982). [14] H. P. Schlickewei and W. M. Schmidt, The number of solutions of polynomialexponential equations, Compositio Math. 120 (2000), 193–225.
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[15] T. N. Shorey and C. L. Stewart, On the Diophantine equation ax2t + bxt y + cy 2 = d and pure powers in recurrence sequences, Math. Scand. 52 (1983), 24–36. [16] N. P. Smart, The algorithmic resolution of Diophantine equations, London Mathematical Society Student Texts Vol. 41, Cambridge University Press, Cambridge, 1998, xvi+243. [17] T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, Cambridge, 1986. [18] B. M. M. de Weger Products of prime powers in binary recurrence sequences II. The elliptic case, with an application to a mixed quadratic-exponential equation, Math. Comp. 47 (1986), 729–739.
´rczes, L. Hajdu, I. Pink A. Be University of Debrecen, Institute of Mathematics H-4002 Debrecen, P.O. Box 400. Hungary S. S. Rout Institute of Mathematics & Applications, Andharua, Bhubaneswar-751029, Odisha, India E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]