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# An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II

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## Abstract

In this paper we study linear forms with rational integer coefficients? (, ), where the are algebraic numbers satisfying the so-called strong independence condition. In standard notation, we prove an explicit estimate of the form Its novel feature is that it contains no factors of the form?.

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... where R = rad(abc), p ′ = min{P(a), P(b), P(c)} and κ is an effective constant. The bound (1.2) becomes subexponential on R only when p ′ is considerably smaller than R, namely, when p ′ < R o (1) with o(1) a function converging to 0. This last condition is rather restrictive and it depends on the prime factorization of a, b, and c. See also Corollary 7 in [5] for a similar result. ...
... About the proofs. The main Diophantine approximation input in our arguments is a result of Matveev [1] concerning bounds for non-vanishing linear forms in complex logarithms. This will allow us to prove Theorem 1.1 in Section 3. Theorem 1.6 is proved in Section 4. It is deduced from Theorem 1.1 via a geometric construction. ...
... We remark that, in our notation, Corollary 2.3 in [1] has the factor (n + 1) 4.5 · 30 n instead of e 7n , but it is an elementary observation that (n + 1) 4.5 · 30 n < e 7n for n ≥ 1. Let A 0 = max{d · h(β), | log β|, 0.16} which is a number that only depends on α in an effective way, and for 1 ≤ j ≤ n define A j = d · log p j . ...
Preprint
We prove a Diophantine approximation inequality for rational points in varieties of any dimension, in the direction of Vojta's conjecture with truncated counting functions. Our results also provide a bound towards the $abc$ conjecture which in several cases is subexponential. The main theorem gives a lower bound for the truncated counting function relative to a divisor with sufficiently many components, in terms of the proximity to an algebraic point. Our methods are based on the theory of linear forms in logarithms and a geometric construction.
... This is A000931 sequence in [16]. Its first few terms are 0, 1, 1, 1, 2, 2, 3,4,5,7,9,12,16,21,28,37,49,65,86,114,151, . . . ...
... (3, 1, 0, 0), (5, 1, 3, 0), (5, 2, 0, 1), (6, 1, 5, 0), (6, 2, 0, 0), (6, 2, 3, 1), (7, 1, 6, 0), (7, 2, 3, 0), (7, 2, 5, 1), (7, 3, 0, 2), (8, 1, 7, 0), (8,2,5,0), (8,2,6,1), (8,3,3,2), (9, 2, 7, 0), (9, 2, 8, 1), (9, 3, 0, 0), (9, 3, 3, 1), (9, 3, 6, 2), (10, 2, 9, 1), (10, 3, 5, 0), (10, 3, 6, 1), (10,3,8,2), (10,4,3,3), (11, 2, 10, 0), (11,3,8,0), (11,4, 0, 2), (11,4,7,3), (12, 3, 10, 0), (12,3,11,2), (12,4,3,0), (12,4,5,1), (12,4,7,2), (12,5,0,4), (13,4,9,1), (13,4,10,2), (13,5,8,4), (14,3,13,0), (14,4,12,2), (14, 5, 0, 2), (14,5,7,3), (14,5,11,4), (15,5,9,1), (15,5,10,2), (15,5,13,4), (15,6,8,5), (16,4,15,2), (16,5,13,2), (16,6,3,4), (17,5,15,2), (17,5,16,4), (17,6,5,0), (17,6,6,1), (17,6,8,2), ( ...
... (3, 1, 0, 0), (5, 1, 3, 0), (5, 2, 0, 1), (6, 1, 5, 0), (6, 2, 0, 0), (6, 2, 3, 1), (7, 1, 6, 0), (7, 2, 3, 0), (7, 2, 5, 1), (7, 3, 0, 2), (8, 1, 7, 0), (8,2,5,0), (8,2,6,1), (8,3,3,2), (9, 2, 7, 0), (9, 2, 8, 1), (9, 3, 0, 0), (9, 3, 3, 1), (9, 3, 6, 2), (10, 2, 9, 1), (10, 3, 5, 0), (10, 3, 6, 1), (10,3,8,2), (10,4,3,3), (11, 2, 10, 0), (11,3,8,0), (11,4, 0, 2), (11,4,7,3), (12, 3, 10, 0), (12,3,11,2), (12,4,3,0), (12,4,5,1), (12,4,7,2), (12,5,0,4), (13,4,9,1), (13,4,10,2), (13,5,8,4), (14,3,13,0), (14,4,12,2), (14, 5, 0, 2), (14,5,7,3), (14,5,11,4), (15,5,9,1), (15,5,10,2), (15,5,13,4), (15,6,8,5), (16,4,15,2), (16,5,13,2), (16,6,3,4), (17,5,15,2), (17,5,16,4), (17,6,5,0), (17,6,6,1), (17,6,8,2), ( ...
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Let (Pn)n≥0 be the Padovan sequence given by P0 = 0, P1 = P2 = 1 and the recurrence formula Pn+3 = Pn+1 + Pn for all n ≥ 0. In this note we study and completely solve the Diophantine equation Pn - 2m = Pn1 - 2m1 in non-negative integers (n, m, n1, m1).
... There are many such in the literature like that of Baker and Wüstholz from [2]. We use the one of Matveev from [15]. Matveev [15] proved the following theorem, which is one of our main tools in this paper. ...
... We use the one of Matveev from [15]. Matveev [15] proved the following theorem, which is one of our main tools in this paper. ...
... which implies that min{m i , l j } < 3 × 10 15 (1 + log(2m 4 )) 2 (1 + log(2m 2 4 )). ...
Preprint
Let $\{P_m\}_{m\ge 0}$ be the sequence of Pell numbers given by $P_0=0, ~ P_1=1$ and $P_{m+2}=2P_{m+1}+P_m$ for all $m\ge 0$. In this paper, for an integer $d\geq 2$ which is square free, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^{2}-dy^{2} =\pm 1$ which is a product of two Pell numbers.
... In order to prove our main result Theorem 1, we first give a general lower bound for linear forms in logarithms due to Matveev [23]. Let η be an algebraic number of degree d with minimal primitive polynomial over the integers ...
... With this notation, Matveev (see [23]) proved the following deep theorem. ...
... Clearly τ is an irrational number. We put M := 1.552 × 10 625 which is an upper bound for 2m according to (23). It then follows from Lemma 2, applied to inequality (30), that ...
Article
The k-Fibonacci sequence {Fn(k)}n starts with the values 0,…,0,1 (a total of k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all integers c having at least two representations as a difference between a k-Fibonacci number and a Pell number. This paper continues and extends the previous work of [J.J. Bravo, C.A. Gómez, and J.L. Herrera, On the intersection of k-Fibonacci and Pell numbers, Bull. Korean Math. Soc. 56(2) (2019), pp. 535–547; S. Hernández, F. Luca, and L.M. Rivera, On Pillai's problem with the Fibonacci and Pell sequences, Soc. Mat. Mex. 25 (2019), pp. 495–507 and M.O. Hernane, F. Luca, S.E. Rihane, and A. Togbé, On Pillai's problem with Pell numbers and powers of 2, Hardy- Ramanujan J. 41 (2018), pp. 22–31].
... We start by recalling the result of Bugeaud, Mignotte, and Siksek ([3], Theorem 9.4, pp. 989), which is a modified version of the result of Matveev [19]. This result is one of our main tools in this paper. ...
... With the above notation, according to Bugeaud, Mignotte, and Siksek [3], Matveev [19] proved the following theorem. ...
... Now, by comparing with (19), we get that (20) (n − n 1 ) log α < 1.92 × 10 26 (1 + log n) 2 . ...
Article
Let { P n } n ≥0 be the sequence of Padovan numbers defined by P 0 = 0, P 1 = 1, P 2 = 1, and P n +3 = P n +1 + P n for all n ≥ 0. In this paper, we find all integers c admitting at least two representations as a difference between a Padovan number and a power of 3.
... Before getting into details, we give a brief description of our method. We first use Matveev's result [17] on linear forms in logarithms to obtain an upper bound for x in terms of m. When m is small, we use Dujella and Pethö's result [6] to decrease the range of possible values that allow us to treat our problem computationally. ...
... In order to prove our main result, we use a few times a Baker-type lower bound for a non-zero linear forms in logarithms of algebraic numbers. We state a result of Matveev [17] about the general lower bound for linear forms in logarithms, but first, recall some basic notations from algebraic number theory. ...
... With the established notations, Matveev (see [17] or [2, Theorem 9.4]), proved the ensuing result. ...
Preprint
In this paper, we explicitly find all solutions of the title Diophantine equation, using lower bounds for linear forms in logarithms and properties of continued fractions. Further, we use a version of the Baker-Davenport reduction method in Diophantine approximation, due to Dujella and Peth\"o. This paper extends the previous work of \cite{Patel}.
... Proof. This is a very slight simplification of Theorem 2.1 of [18] applied with n = 3. Our only change is to note that |b j | A j /A 1 ≥ 1 for j = 1, so the outer max in Matveev's inequality B ≥ max {1, max {|b j | A j /A 1 : 1 ≤ j ≤ 3}} is not needed. ...
... It is because the log α j 's are Q-linearly independent in both of these cases that we can use Theorem 2.1 of [18] in this work. ...
... It is this worse dependence on log B than in the non-degenerate case that leads to the degenerate case having an impact on the results obtained in practice. Fortunately, it is the constants that are important and our estimates should lead to good results when compared to published previously ones (e.g., [18]). See the example in the next section for evidence of this. ...
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In this paper, we provide a technique to obtain explicit bounds for problems that can be reduced to linear forms in three logarithms of algebraic numbers. This technique can produce bounds significantly better than general results on lower bounds for linear forms in logarithms. We also give worked examples to demonstrate the improvements and to help readers see how to apply our technique.
... Many Diophantine equations can be solved by reducing them to an instance in which one can apply lower bounds for linear forms in logarithms of algebraic numbers. Now we give a theorem deduced from Corollary 2.3 of Matveev [20] and provides a large upper bound for the subscript n in the equation (2) (also see Theorem 9.4 in [12]). ...
... Substituting this upper bound on k into (24), we obtain n < 2.67 · 10 30 · 680 8 · (log 680) 5 < 1.45 · 10 57 . Now, let us apply Lemma 4 to (20). For this, let ...
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Let k≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 2$$\end{document} and let (Pn(k))n≥2-k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_{n}^{(k)})_{n\ge 2-k}$$\end{document} be the k-generalized Pell sequence defined by Pn(k)=2Pn-1(k)+Pn-2(k)+⋯+Pn-k(k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned} P_{n}^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)} \end{aligned}\end{document}for n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document} with initial conditions P-(k-2)(k)=P-(k-3)(k)=⋯=P-1(k)=P0(k)=0,P1(k)=1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned} P_{-(k-2)}^{(k)}=P_{-(k-3)}^{(k)}=\cdots =P_{-1}^{(k)}=P_{0}^{(k)}=0,P_{1}^{(k)}=1. \end{aligned}\end{document}In this paper, we show that 12, 13, 29, 33, 34, 70, 84, 88, 89, 228, and 233 are the only k-generalized Pell numbers, which are concatenation of two repdigits with at least two digits.
... There are many such in the literature like that of Baker and Wüstholz from [2]. We use the one of Matveev from [14]. Matveev [14] proved the following theorem, which is one of our main tools in this paper. ...
... We use the one of Matveev from [14]. Matveev [14] proved the following theorem, which is one of our main tools in this paper. ...
Article
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Consider the sequence $$\{{F}_{n}\}_{n \ge 0}$$ of Fibonacci numbers defined by $${F}_0=0$$, $${F}_1 =1$$, and $${F}_{{n}+2}= {F}_{{n}+1}+ {F}_{n}$$ for all $$n\ge 0$$. In this paper, we find all integers c having at least two representations as a difference between a Fibonacci number and a power of 3.
... Since the proof of the following lemma is given firstly in [9] and then in [3], we will omit its proof. ...
... Lemma 5 [3,9]. Let L be a number field of degree D and α 1 , α 2 , . . . ...
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In this paper, we find non-negative (n, m, a) integer solutions of the diophantine equation $$F_{n}-F_{m}=3^{a}$$, where $$F_{n}$$ and $$F_{m}$$ are Fibonacci numbers. For proving our theorem, we use lower bounds in linear forms.
... Let us now work a little bit on (15) in order to find an improved upper bound on k − t. ...
... Therefore, (15) implies that ...
Preprint
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Let (L k) k≥0 be the Lucas sequence given by L 0 = 2, L 1 = 1 and L k+2 = L k+1 + L k for k ≥ 0. In this paper, we are interested in finding all powers of two which are sums of three Lucas numbers, i.e., we study the Diophantine equation L k + L l + L t = 2 d in nonnegative integers k, l, t and d. The proof of our main theorem uses lower bounds for linear forms in logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in diophantine approximation. This paper continues and extends the previous work of J.J. Bravo and F. Luca [3] and our previous work [9].
... Let {L n } n≥0 be the sequence of Lucas numbers given by L 0 = 2, L 1 = 1 and L n+2 = L n+1 + L n for all n ≥ 0. This is sequence A000032 on the Online Encyclopedia of Integer Sequences (OEIS). The first few terms of this sequence are {L n } n≥0 = 2, 1, 3,4,7,11,18,29,47,76,123,199,322,521,843,1364, 2207, 3571, . . . . ...
... In particular, if η = p/q is a rational number with gcd(p, q) = 1 and q > 0, then h(η) = log max{|p|, q}. The following are some of the properties of the logarithmic height function h(·), which will be used in the next sections of this paper without reference: [18], which is one of our main tools in this paper. ...
... Projection is often necessary because Lipschitz continuity of f is assumed in (QSA2). This requires Γ to be Lipschitz continuous when employing 1qSGD (21) and ∇Γ when using 2qSGD (22) [11]. Restricting {Θ t } to a closed and bounded set is a way to relax these requirements. ...
... Bounds on C have been refined over many years. Unfortunately, current bounds on this constant grow rapidly with d, such as the doubly exponential bounds obtained in [21] and [22]. Many extensions are possible-we do not require that each frequency is the logarithm of a rational number-see [10] for the current state of the art. ...
Preprint
Stochastic approximation is a foundation for many algorithms found in machine learning and optimization. It is in general slow to converge: the mean square error vanishes as $O(n^{-1})$. A deterministic counterpart known as quasi-stochastic approximation (QSA) is a viable alternative in many applications, including gradient free optimization and reinforcement learning. It was assumed in recent prior research that the optimal achievable convergence rate is $O(n^{-2})$. It is shown in this paper that through design it is possible to obtain far faster convergence, of order $O(n^{-4+\delta})$, with $\delta>0$ arbitrary. Two acceleration techniques are introduced for the first time to achieve this rate of convergence. The theory is also specialized within the context of gradient-free optimization, and tested on standard benchmarks. The main results are based on a combination of recent results from number theory and techniques adapted from stochastic approximation theory.
... The following consequence of Corollary 2.3 of Matveev [9] provides a large upper bound for the subscript n in (1.2) (see also [5,Theorem 9.4]). ...
... 3.1. The only solutions (n, m, a) of the Diophantine equation (1.2) in nonnegative integers m < n and a are (1, 0, 0), (2, 0, 0), (3, 0, 1), (6, 0, 3), (3, 1, 0), (4, 1, 1), (5, 1, 2), (3, 2, 0), (4, 3, 0), (4, 2, 1), (5, 2, 2),(9,3,5), (5, 4, 1),(7,5,3),(8,5,4),(8,7,3). ...
Article
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We solve the Diophantine equation in the tittle in nonnegative integers m,n, and a. In order to prove our result, we use lower bounds for linear forms in logarithms and a version of the Baker–Davenport reduction method in Diophantine approximation.
... The key tool for the proofs of Theorems 1.2 to 1.5 is the following immediate consequence of a theorem of Matveev [7]. The height h of an algebraic number is defined in (2.1). ...
Preprint
Among other results, we establish, in a quantitative form, that any sufficiently large integer cannot simultaneously be divisible only by very small primes and have very few digits in its Zeckendorf representation.
... In addition, there is another explicit form of Baker's theory due to Matveev [17,Corollary 2.3]. However, for our purpose the one of Baker and Wüstholz [3] gives a slightly better result. ...
Preprint
Recently E.~Bombieri and N.~M.~Katz (2010) have demonstrated that several well-known results about the distribution of values of linear recurrence sequences lead to interesting statements for Frobenius traces of algebraic curves. Here we continue this line of study and establish the M\"obius randomness law quantitatively for the normalised form of Frobenius traces.
... The next result is due to Matveev [22] and can be used to get a better lower bound on the linear form (22) than (23). Theorem 5.3 (Matveev). ...
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In this paper we give an upper bound on the number of extensions of a triple to a quadruple for the Diophantine m-tuples with the property D(4) and confirm the conjecture of uniqueness of such extension in some special cases.
... For the proofs of (10) and further details, we refer the reader to the book of Bombieri and Gubler [1]. We recall the result of Bugeaud, Mignotte, and Siksek [2, Theorem 9.4, p. 989], which is a modified version of the result of Matveev [10], which is one of our main tools in this paper. ...
Preprint
Let $\{P_{n}\}_{n\geq 0}$ be the sequence of Padovan numbers defined by $P_0=0$, $P_1 =1=P_2$ and $P_{n+3}= P_{n+1} +P_n$ for all $n\geq 0$. In this paper, we find all repdigits in base $10$ which can be written as a sum of three Padovan numbers.
... Since a set of 6 elements admits 10 partitions of signature (3,3) and 15 partitions of signature (4,2), the total number of suitable pairs is bounded by 10(κ(3) + κ(3)) + 15(κ(4) + κ(2)) + κ(6) = 20κ(3) + 15κ(4) + κ(6) + 15. ...
Preprint
We show that, apart from some obvious exceptions, the number of trinomials vanishing at given complex numbers is bounded by an absolute constant. When the numbers are algebraic, we also bound effectively the degrees and the heights of these trinomials.
... The main powerful tool to prove Theorem 1 is a lower bound for a linear form logarithms à la Baker, which was given by the following result of Matveev (see [21] or [2] (Theorem 9.4)). ...
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The k-generalized Fibonacci sequence ( F n ( k ) ) n (sometimes also called k-bonacci or k-step Fibonacci sequence), with k ≥ 2 , is defined by the values 0 , 0 , … , 0 , 1 of starting k its terms and such way that each term afterwards is the sum of the k preceding terms. This paper is devoted to the proof of the fact that the Diophantine equation F m ( k ) = m t , with t > 1 and m > k + 1 , has only solutions F 12 ( 2 ) = 12 2 and F 9 ( 3 ) = 9 2 .
... The following theorem is deduced from Corollary 2.3 of Matveev [10] and provides a large upper bound for the subscripts n and m in the equation (4)(also see [3,Theorem 9.4]). ...
Article
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Let Pn, Qn, Bn, and Cn denote, respectively Pell, Pell-Lucas, balancing , and Lucas-balancing numbers. In this study, we show that if PmPn is a repdigit, then PmPn ∈ {0, 1, 2, 4, 5} and that if QmQn is a repdigit, then QmQn = 4. Moreover , we show that if BmBn is a repdigit, then BmBn ∈ {0, 1, 6} and that if CmCn is a repdigit, then CmCn ∈ {1, 3, 9, 99}.
... The following theorem is deduced from Corollary 2.3 of Matveev [8] and provides a large upper bound for the subscript í µí± in the equations (1.4) and (1.5)(also see Theorem 9.4 in [2]). ...
... We shall also need the following general lower bound for linear forms in logarithms due to Matveev [14] (see also the paper of Bugeaud, Mignotte, and Siksek [5, Theorem 9.4]). ...
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Let (L k) k≥0 be the Lucas sequence given by L 0 = 2, L 1 = 1 and L k+2 = L k+1 + L k for k ≥ 0. In this note, we find all positive integer solutions of the Diophantine equation L k − L l = 2 t in nonnegative integers k, l, and t. The proof of our main theorem uses lower bounds for linear forms in logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in diophantine approximation. This note continues and extends the previous work of J.J. Bravo and F. Luca [3].
... Firstly, we discuss a lower bound for linear forms in logarithms due to Bugeaud, Mignotte, and Siksek [2], which is a consequence of the result of Matveev [10]. ...
... We recall the result of Bugeaud, Mignotte, and Siksek [2, Theorem 9.4, p. 989], which is a modified version of the result of Matveev [10], which is one of our main tools in this paper. ...
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Let $$\{P_{n}\}_{n\ge 0}$$ be the sequence of Padovan numbers defined by $$P_0=0$$, $$P_1 =1=P_2$$, and $$P_{n+3}= P_{n+1} +P_n$$ for all $$n\ge 0$$. In this paper, we find all repdigits in base 10 which can be written as a sum of three Padovan numbers.
... We recall the result of Bugeaud, Mignotte, and Siksek ([2], Theorem 9.4, pp. 989), which is a modified version of the result of Matveev [8], which is one of our main tools in this paper. ...
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Let { B n } n ≥0 be the sequence of Balancing numbers defined by B0 = 0, B1 = 1, and Bn +2 = 6 Bn +1 − B n for all n ≥ 0. In this paper, we find all repdigits in base 10 which can be written as a sum of three Balancing numbers.
... • h(α l ) = |l|h(α), l ∈ Z. One of the main tools in this paper is a widely used estimate on lower bounds for linear forms in complex logarithms due to Matveev [8]. ...
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In this paper, we find all sums of two Fibonacci numbers which are close to a power of 2. As a corollary, we also determine all Lucas numbers close to a power of 2. The main tools used in this work are lower bounds for linear forms in logarithms due to Matveev and Dujella-Peth\"{o} version of the Baker-Davenport reduction method in diophantine approximation. This paper continues and extends the previous work of Chern and Cui.
... The following lemma is deduced from Corollary 2.3 of Matveev (see [8]). Lemma 2.1. ...
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The sequence of balancing numbers $(B_n)$ is defined by the recurrence relation $B_n=6B_{n-1}-B_{n-2}$ for $n\geq2$ with initial conditions $B_0=0$ and $B_1=1.$ $B_n$ is called the $n$th balancing number. In this paper, we find all repdigits in the base $b,$ which are sums of four balancing numbers. As a result of our theorem, we state that if $B_n$ is repdigit in the base $b$ and has at least two digits, then $(n,b)=(2,5),(3,6)$. Namely, $B_2=6=(11)_5$ and $B_3=35=(55)_6. ... We need the following theorem from the theory of lower bounds on linear forms in logarithms of algebraic numbers. Recall Theorem 9.4 of [3], which is a modified version of a result of Matveev [7]. Let L be an algebraic number field of degree d L and let η 1 , η 2 , . . . ... Preprint Full-text available Consider the diophantine equation$(3^{x_1}-1)(3^{x_2}-1)=(5^{y_1}-1)(5^{y_2}-1)$in positive integers$x_1\le x_2$, and$y_1\le y_2$. Each side of the equation is a product of two terms of a given binary recurrence, respectively. In this paper, we prove that the only solution to the title equation is$(x_1,x_2,y_1,y_2)=(1,2,1,1). The main novelty of our result is that we allow products of two terms on both sides. ... The following theorem is deduced from Corollary 2.3 of Matveev [12] and provides a large upper bound for the subscripts in the equation (5) (also see Theorem 9.4 in [3]). ... Preprint In this paper, we will answer the question of when the sum or the difference of x-th powers of any two Fibonacci numbers becomes a Fibonacci number or a Lucas number. We prove that if this is possible with x ≥ 3, then x may be only 3, 4, 5, or 10. Also, we give the result that the Diophantine equation in the title with x ≥ 3 has no solutions for n−m > 2. ... denote effectively computable positive numbers. This is a linear form in two logarithms and by [10], [2] or [15] we see from (20) and (21), since λ 2 /λ 1 is not a root of unity, that (22) log |ζ(λ 2 /λ 1 ) n − 1| > −c 2 log(n + 1) log max(4, A) ... Preprint We give estimates from below for the greatest prime factor of the n-th term of a binary recurrence sequence. ... The following theorem, which is deduced from Corollary 2.3 of Matveev [9], provides a large upper bound for the subscript m 1 in the equation (1.3) (also see Theorem 9.4 in [5]). ... Article Full-text available Let (Ln) be the Lucas sequence deÖned by Ln = Ln1 +Ln2 for n � 2 with initial conditions L0 = 2 and L1 = 1: A repdigit is a nonnegative integer whose digits are all equal. In this paper, we show that if Ln + Lm is a repdigit, then Ln + Lm = 2; 3; 4; 5; 6; 7; 8; 9; 11; 22; 33; 77; 33 Article Let P_k and Q_k be the k-th terms of the Pell and Pell-Lucas sequences respectively. In this paper, we find all the solutions of the Diophantine equations P_{n+1}^x±P_{n−1}^x=Q_m in nonnegative integer variables (m, n, x). Article We study the extendibility of a $$D(-1)$$-pair {1, p}, where p is a Fermat prime, to a $$D(-1)$$-quadruple in $$\mathbb{Z} [\sqrt{-t}], t > 0$$. Article In this paper we find (n, m, a) solutions of the Diophantine equation $$L_{n}-L_{m}=2\cdot 3^{a}$$, where $$L_{n}$$ and $$L_{m}$$ are Lucas numbers with $$a\ge 0$$ and $$n>m\ge 0$$. For proving our theorem, we use lower bounds for linear forms in logarithms and Baker–Davenport reduction method in Diophantine approximation. Article The k–generalized Pell sequence P(k):=(Pn(k))n≥-(k-2) is the linear recurrence sequence of order k, whose first k terms are 0,…,0,1 and satisfies the relation Pn(k)=2Pn-1(k)+Pn-2(k)+⋯+Pn-k(k), for all n,k≥2. In this paper, we investigate about integers that have at least two representations as a difference between a k–Pell number and a perfect power. In order to exhibit a solution method when b is known, we find all the integers c that have at least two representations of the form Pn(k)-bm for b∈[2,10]. This paper extends the previous works in Ddamulira et al. (Proc. Math. Sci. 127: 411–421, 2017) and Erazo et al. (J. Number Theory 203: 294–309, 2019). Preprint Full-text available In this study, we search for Pell and Pell-Lucas numbers, which are con-catenations of two repdigits and …nd these numbers to be only 12; 29; 70 and 14; 34; 82 respectively. We use the Baker-Davenport reduciton method while …nding the solutions. Preprint Full-text available Padovan numbers are given by P k = P k 2 + P k 3 where (P0; P1; P2) = (0; 1; 1) for k 2 Z; k 3 and positive integers with all digits equal are called repdigit. In this paper, the Padovan numbers, which are the di¤er-ence of two repdigits, were examined and found to be P k Chapter Gelfond, by a refinement of his method, obtained a positive lower bound for the absolute value of $$\beta _1 \log \alpha _1+\beta _2\log \alpha _2$$ where $$\beta _1,\beta _2$$ denote algebraic numbers not both 0, and $$\alpha _1,\alpha _2$$ denote algebraic numbers not 0 or 1, with $$\log \alpha _1/\log \alpha _2$$ irrational. Gelfond also remarked that an analogous theorem for linear forms in arbitrarily many logarithms of algebraic numbers would be of great value for the solution of some apparently very difficult problems in number theory. In 1966–68, Baker established such a result. See his papers [1] and his prize winning book [2]. Corollaries 7.2.1, 7.2.2 and 7.2.3 resolve the multidimensional analogue of Hilbert’s seventh problem. We have chosen to give as applications, some results on Pillai’s equation, the growth of the greatest prime factor of polynomial values and effective version of Thue’ theorem; see Sects. 7.3 and 7.4. Article In this paper, we study the titular Diophantine equation for a fixed positive integer y ≥ 3 in nonnegative integers m, n, and a. We show that the nonnegative integer solutions (n, m, a) are finite in number, and we provide a bound for them. Chapter We begin with some basic tools necessary for the proof of Theorem 7.1.1 in Sect. 8.1. First, Theorem 7.1.1 is reduced to an equivalent statement; see Theorem 8.1.2. In Sect. 8.1.1, we derive a simple, but useful, non-trivial lower bound for a non-vanishing linear form in logarithms of algebraic numbers with bounded coefficients. Section 8.1.2 provides construction of an augmentative polynomial. In Sect. 8.1.3, we give the construction of the auxiliary polynomial $$\Phi (Z_0,\ldots ,Z_{n-1})$$ in several variables which generalises the function of a single complex variable employed by Gelfond. Basic estimates on $$\Phi$$ are shown in Sect. 8.1.4. The main difficulty is in the interpolation techniques. Usually the order of the derivatives is increased while leaving the points of interpolation fixed. Baker used a special extrapolation procedure in which the range of interpolation points is extended while the order of the derivatives is reduced, and the absolute values of these derivatives are shown to be very small. See Sects. 8.1.5 and 8.1.6. Article Full-text available In this study, we show that if $$2\le m\le n$$ and $$F_{m}F_{n}$$ represents a repdigit, then (m, n) belongs to the set \begin{aligned} \left\{ (2,2),(2,3),(3,3),(2,4),(3,4),(4,4),(2,5),(2,6),(2,10)\right\} \text {.} \end{aligned}Also, we show that if $$0\le m\le n$$ and $$L_{m}L_{n}$$ represents a repdigit, then (m, n) belongs to the set\begin{aligned} \left\{ \begin{array}{c} (0,0),(0,1),(1,1),(0,2),(1,2),(2,2),(0,3), \\ (1,3),(1,4),(1,5),(2,5),(3,5),(4,5) \end{array} \right\} \text {.} \end{aligned} Article In this paper, we determine all Lucas numbers that are sums of two repdigits. The largest one is L14 = 843 = 66 + 777. Article Full-text available Let $$\{X_\ell \}_{\ell \ge 1}$$ be the sequence of X-coordinates of the positive integer solutions (X, Y) of the Pell equation $$X^2-dY^2=\pm 1$$ corresponding to a nonsquare integer $$d>1$$. We show that there are only a finite number of nonsquare integers $$d > 1$$ such that there are at least two different elements of the sequence $$\{X_\ell \}_{\ell \ge 1}$$ that can be represented as a linear combination of prime powers with fixed primes and coefficients, restricted to the condition that the exponent of the largest prime is the greatest of all exponents. Moreover, we solve explicitly the case in which two of the X-coordinates above are a sum of a power of two and a power of three under the above condition on the exponents. This work is motivated by the recent paper Bertók et al. (Int J Number Theory 13(02):261–271, 2017). Preprint Full-text available Letr\ge 1$be an integer and${\bf U}:=(U_{n})_{n\ge 0} $be the Lucas sequence given by$U_0=0$,$U_1=1, $and$U_{n+2}=rU_{n+1}+U_n$, for all$ n\ge 0 $. In this paper, we show that there are no positive integers$r\ge 3,~x\ne 2,~n\ge 1$such that$U_n^x+U_{n+1}^x$is a member of${\bf U}$. Preprint Full-text available In this paper, we give a constant$C$in \cite[Theorem 1.2]{sha2014bounding} by using an explicit Baker's inequality, hence we have an explicit bound of the integral points on modular curves. Article Diophantine tuples are sets of positive integers with the property that the product of any two elements in the set increased by the unity is a square. In the main theorem of this paper it is shown that any Diophantine triple, the second largest element of which is between the square and four times the square of the smallest one, is uniquely extended to a Diophantine quadruple by joining an element exceeding the largest element in the triple. A similar result is obtained under the hypothesis that the two smallest elements have the form T2+2T, 4T4+8T3−4T for some positive integer T, which we encounter as an exceptional case. The main theorem implies that the same is valid for triples with smallest elements KA2, 4KA4±4A for some positive integers A and K∈{1,2,3,4}. Preprint Full-text available Let$(F_n)_{n\geq 0}$and$(L_n)_{n\geq 0}$be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of$ a $and$ b $, we mean the both concatenations$\overline{ab}$and$\overline{ba}$together, where$ a $and$ b $are any two non negative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations$ F_n=10^d F_m +L_k $and$ F_n=10^d L_m+F_k $in non-negative integers$ (n,m,k) ,$where$ d $denotes the number of digits of$ L_k $and$ F_k \$, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.
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Resume On sait que la méthode classique de Schneider (en une variable) permet de minorer des combinaisons linéaires de deux logarithmes de nombres algébriques avec des coefficients algébriques. Nous généralisons cette méthode en plusieurs variables pour minorer des combinaisons linéaires de plusieurs logarithmes.
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A theorem on the successive minima of lattices corresponding to the integer solutions of systems of linear equations is proved. As a corollary, theorems on the successive minima are obtained for the set of solutions of equations of the form for fixed in an algebraic number field and for variable equal either to 1 or a root of unity.Bibliography: 13 titles.
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For certain number theoretical applications, it is useful to actually compute the effectively computable constant which appears in Baker's inequality for linear forms in logarithms. In this note, we carry out such a detailed computation, obtaining bounds which are the best known and, in some respects, the best possible. We show inter alia that if the algebraic numbers α 1 , …, α n all lie in an algebraic number field of degree D and satisfy a certain independence condition, then for some n 0 ( D ) which is explicitly computed, the inequalities (in the standard notation) have no solution in rational integers b 1 , …, b n ( b n ≠ 0) of absolute value at most B , whenever n ≥ n 0 ( D ). The very favourable dependence on n is particularly useful.
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Let α 1 , …, α n an be non-zero algebraic numbers with degrees at most d and heights respectively A l , …, A n (all A j ≥ 4) and let b 1 , …, b n be rational integers with absolute values at most B (≥ 4). Denote by p a prime ideal of the field and suppose that p divides the rational prime p . Write Then it is shown that for some effectively computable constant C > 0 depending only on n, d and p . The argument suffices to prove similarly that in the complex case, if for any fixed determination of the logarithms, then for some effectively computable constant C ′ > 0 depending only on n and d (and he determination of the logarithms).
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For every algebraic number of degree there exist effective positive constants and such that for any rational integers and we have
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Siegel's lemma in its simplest form is a statement about the existence of small-size solutions to a system of linear equations with integer coefficients: such results were originally motivated by their applications in transcendence. A modern version of this classical theorem guarantees the existence of a whole basis of small "size" for a vector space over a global field (that is number field, function field, or their algebraic closures). The role of size is played by a height function, an important tool from Diophantine geometry, which measures "arithmetic complexity" of points. For many applications it is also important to have a version of Siegel's lemma with some additional algebraic conditions placed on points in question. I will discuss the classical versions of Siegel's lemma, along with my recent results on existence of points of bounded height in a vector space outside of a finite union of varieties over a global field.