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An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II
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?.
... Then we get Q m+1 − Q m = Q m + Q m−1 . Thus by Lemma 5, we get the solution (m, m − 1, d, k, Q m+1 − Q m ) = (2,1,8,1,8), which gives the solution (n, m, d, k, Q n − Q m ) = (3,2,8,1,8). From now on, assume that n ≥ 100, m ≥ 1 and n − m ≥ 2. Since Q n is even for all n, Q n − Q m is even. ...
... Then we get Q m+1 − Q m = Q m + Q m−1 . Thus by Lemma 5, we get the solution (m, m − 1, d, k, Q m+1 − Q m ) = (2,1,8,1,8), which gives the solution (n, m, d, k, Q n − Q m ) = (3,2,8,1,8). From now on, assume that n ≥ 100, m ≥ 1 and n − m ≥ 2. Since Q n is even for all n, Q n − Q m is even. ...
... Then we get Q m+1 − Q m = Q m + Q m−1 . Thus by Lemma 5, we get the solution (m, m − 1, d, k, Q m+1 − Q m ) = (2,1,8,1,8), which gives the solution (n, m, d, k, Q n − Q m ) = (3,2,8,1,8). From now on, assume that n ≥ 100, m ≥ 1 and n − m ≥ 2. Since Q n is even for all n, Q n − Q m is even. ...
In this paper, we determine all repdigits, which are difference of two Pell and Pell-Lucas numbers. It is shown that the largest repdigit which is difference of two Pell numbers is 99 = 169 − 70 = P 7 − P 6 and the largest repdigit which is difference of two Pell-Lucas numbers is 444 = 478 − 34 = Q 7 − Q 4 .
... See the sequence A000931 in the OEIS. The first few terms of Padovan sequence are 1, 1, 1, 2, 2, 3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200, . . . . The Perrin sequence {E n } n≥0 is given by E 0 = 3, E 1 = 0, E 2 = 2, and E n+3 = E n+1 + E n , for n ≥ 0 . ...
... The first tool we need is the following result due to Matveev [7]. Here we use the version of Bugeaud, Mignotte and Siksek [2, Theorem 9.4]. ...
... The only solutions (m, n, k) of the Diophantine equation P m = P n P k (5.13) in non-negative integers m, n and k with 1 ≤ n ≤ k are (0, 1, 1), (1, 1, 1), (2, 1, 1), (3, 1, 2), (4, 1, 2), (6, 2, 2), (7,1,3), and (10, 1, 4). ...
... We will have need of bounds for linear forms in complex and p-adic logarithms; results of Matveev [15] and Yu [25] are applicable in the broadest generality. ...
... Theorem 4 (Matveev, 2000). Let α 1 , . . . ...
... If γ ∈ {5, 7}, we thus have ν 2 (3 u γ 1 − γ 2 ) bounded above by 72.2 log 3 9(log 2) 3 max log Combining this with (15) and using that u > 10 5 , we find that 17 log 2 u log γ < 72.2 log 3 9(log 2) 3 log u 3 log 2 + 1 log 3 + log(3 log 2) + 0.4 ...
We show how to effectively solve 5-term $S$-unit equations when the set of primes $S$ has cardinality at most 3, and use this to provide an explicit answer to an old question of D.J. Newman on representations of integers as sums of $S$-units.
... , h p q = log max{|p|, q} p q ∈ Q, q > 0, gcd(p, q) = 1 . Next, we give the general lower bound for linear forms in logarithms due to Matveev [11]. Let K be a number field of degree D over Q, let α 1 , . . . ...
... With this notation, the main result of Matveev [11] implies the following estimate. ...
... 11) where now r := (−2 mθ + λ m−n )/π κ := −2θ/π, μ m−n := λ m−n /π, A := 971, and B := ρ 3/2 .Here, m < n < M := 2.4 × 10 20 . This time, applying Lemma 5.1 to inequality (8.11) for each n − m ∈ [1, 367], we find computationally that q 42 = 62878052392513962537203 is the denominator of the first convergent of the continued fraction of κ, such that q 42 > 6M , the minimum value of is > 1.18018 × 10 −3 and the maximum value of log(Aq 42 / )/ log B is 150.In short, we have showed the following.Lemma 8.4. ...
The integer sequence defined by $$P_{n+3}=P_{n+1}+P_{n}$$ P n + 3 = P n + 1 + P n with initial conditions $$P_{0}=P_{1}=P_{2}=1$$ P 0 = P 1 = P 2 = 1 is known as the Padovan sequence $$(P_{n})_{n\in \mathbb {Z}}$$ ( P n ) n ∈ Z . The Perrin sequence $$(R_{m})_{m\in \mathbb {Z}}$$ ( R m ) m ∈ Z satisfies the same recurrence equation as the Padovan sequence but with starting values $$R_{0}=3$$ R 0 = 3 , $$R_{1}=0$$ R 1 = 0 , and $$R_{2}=2$$ R 2 = 2 . In this note, we solve the Diophantine equation $$P_{n}=\pm R_{m}$$ P n = ± R m with $$(n,m)\in \mathbb {Z}^{2}$$ ( n , m ) ∈ Z 2 .
... To obtain precise bounds of linear forms in logarithms is an important part of our argument. The Matveev's result [10] plays a crucial role to get lower bounds for the non-zero linear forms in logarithms. ...
... Theorem 2.1 (Matveev's theorem [10]) Let γ 1 , γ 2 , . . . , γ t be real algebraic numbers, b 1 , b 2 , . . . ...
... From (1.6) and as α < A calculation on Mathematica reveals that the solutions (n, m) of (3.1) are {(5, 0), (9, 0), (3,2), (3,4), (4,4), (4, 2), (5, 3), (6, 2), (6, 4), (7,5), (7,6), (8,7), (9,3), (10,9), (11,2), (11,4), (15, 7)}. ...
Padovan and Perrin sequences are ternary recurrent sequences that satisfy the same relation \( w_n = w_{n-2} + w_{n-3} \) with different initial conditions \( (w_0, w_1, w_2)= (1,1,1) \) and (3, 0, 2) , respectively. In this study we compute all pairs of Padovan and Perrin numbers that are multiplicatively dependent.
... Consequently, Bravo et al. [6] searched for all k-Lucas numbers that are concatenations of two repdigits. Later, Şiar and Keskin [16] showed that 12,13,29,33,34,70,84,88,89,228 and 233 are the only k-generalized Pell numbers, which are concatenations of two repdigits with at least two digits. In this paper, we find all the k-Pell-Lucas numbers that are concatenations of two repdigits. ...
... For the proof of Theorem 1.1, we first find an upper bound for n in terms of k by applying Matveev's result on linear forms in logarithms [12]. When k is small, the theory of continued fractions suffices to lower such bounds and complete the calculations. ...
... With the above notation, Matveev (see [12] or [7, Theorem 9.4]) proved the following result. ...
For any integer $k \geq 2$, let $\{Q_{n}^{(k)} \}_{n \geq -(k-2)}$ denote the $k$-generalized Pell-Lucas sequence which starts with $0, \dots ,2,2$($k$ terms) where each next term is the sum of the $k$ preceding terms. In this paper, we find all the $k$-generalized Pell-Lucas numbers that are concatenations of two repdigits.
... can be viewed as concatenation of three repdigits where a, l, m, k ≥ 1 and 0 ≤ a, b, c ≤ 9. Let {P n } n≥0 be the Padovan sequence given by P n+3 = P n+1 + P n for all n ≥ 1 with P 0 = P 1 = P 2 = 1. The first few terms of this sequence are 1, 1, 1, 2, 2, 3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265,351, . . . . ...
... In [14], Rayguru and Panda showed that 35 is the only balancing number which is concatenation of two repdigits. Recently, Siar and Keskin [13] found that 12,13,29,33,34,70,84,88,89,228, and 233 are the only k-generalized Pell numbers, which are concatenations of two repdigits with at least two digits. Erduvan and Keskin [9], in their work obtained Lucas numbers which are concatenations of two repdigits. ...
... With these notations, Matveev (see [12] or [4, Theorem 9.4]) proved the following result. ...
Padovan sequence is a ternary recurrent sequence defined by the recurrence relation Pn+3 = Pn+1 + Pⁿ with initial terms P0 = P1 = P2 = 1. In this study it is shown that 114, 151, 200, 265, 351, 465, 616, 816, 3329, 4410, 7739, 922111 are the only Padovan numbers which are concatenations of three repdigits.
2020 Mathematics Subject Classification: 11B39; 11J86; 11D61.
... can be viewed as concatenation of three repdigits where a, l, m, k ≥ 1 and 0 ≤ a, b, c ≤ 9. Let {P n } n≥0 be the Padovan sequence given by P n+3 = P n+1 + P n for all n ≥ 1, with P 0 = P 1 = P 2 = 1. The first few terms of this sequence are 1, 1, 1, 2, 2, 3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265,351, . . . . ...
... In [14], Rayguru and Panda showed that 35 is the only balancing number which is concatenation of two repdigits. Recently, Siar and Keskin [13] found that 12,13,29,33,34,70,84,88,89,228, and 233 are the only k-generalized Pell numbers, which are concatenations of two repdigits with at least two digits. Erduvan and Keskin [9], in their work obtained Lucas numbers which are concatenations of two repdigits. ...
... With these notations, Matveev (see [12] or [4, Theorem 9.4]) proved the following result. ...
Padovan sequence is a ternary recurrent sequence defined by the recurrence relation $P_{n+3}=P_{n+1}+P_{n}$ with initial terms $P_{0}=P_{1}=P_{2}=1.$ In this study it is shown that $114,151,200,265,351,465,616,816,3329,4410,7739,922111$ are the only Padovan numbers which are concatenations of three repdigits.
... Diophantus raised the problem of finding four (positive rational) numbers a 1 , a 2 , a 3 , a 4 such that a i a j + 1 is a square for each 1 ≤ i < j ≤ 4 and gave a solution 1 16 , 33 16 , 17 4 , 105 16 . The first set of four positive integers {1, 3, 8, 120} with this property above was found by Fermat. ...
... Section 5 of this paper will be devoted to the proof of our main theorem. For this, we will apply a Matveev [16] result and then we will end by applying the Baker-Davenport reduction method. ...
... So we take In this section, we will use another theorem for the lower bounds of linear forms in logarithms which differs from that in above section and the Baker-Davenport reduction method to deal with the remaining cases. We recall the following result due to Matveev [16] (see also Lemma 10 in [13]). ...
In this paper, we study the extensibility of the D(4)-triple
{1, b, c}, where 1 < b < c, by proving that such a set cannot be extended to an irregular D(4)-quadruple only for some values of c. For this study, we will use the classical methods based on the resolution of the binary recurrence sequences with new approaches in order to confirm a conjecture of uniqueness of such an extension
... The following lemma is owing to Matveev in [11] and also in [3]. Using this lemma, we find a large bound for the n in the equations (3) and (4). ...
... By using (11) and (15), we find n < 1.89 · 10 29 thanks to easy calculation. We apply Lemma 2.2 for reduce the upper bound on n. ...
In this study, we search for Pell and Pell-Lucas numbers, which are concatenations of two repdigits, and find these numbers to be only 12, 29, 70 and 14, 34, 82 respectively. We use the Baker-Davenport basis reduction method while finding the solutions.
... In order to establish (6) we shall appeal to an estimate for linear forms in the logarithms of rational numbers due to Matveev [8], [9]. The upper bound (7) follows from an averaging argument based on a result of Ennola [3]. ...
... There exists an effectively computable positive number c 0 such that log |Λ| > −c n 0 log A 1 ... log A n log B. Proof. This follows from Theorem 2.2 of Nesterenko [10], which is a special case of the work of Matveev [8], [9]. Let x and y be positive real numbers with y ≥ 2 and let Ψ(x, y) denote the number of positive integers of size at most x all of whose prime factors are of size at most y. ...
We show that the difference between consecutive terms in sequences of integers whose greatest prime factor grows slowly tends to infinity.
... Theorem 4 (Matveev's Theorem, [19]) Assume that α 1 , . . . , α t are positive real algebraic numbers in a real algebraic number field K of degree d K , and let b 1 , . . . ...
... Now, we follow the same calculations as we did for (16) to show that the bound n(2) < 1216 is valid. Finally, we write a short computer programme to check that the variables n, m, k, l, a and d are satisfying (1) by using the bounds n(k) for 2 ≤ k ≤ 650 together with (19) and (8). As a result, we find that there is no new solution of (1) except for those that were given in Theorem 1. ...
Let $ k \geq 2 $ and let $ ( L_{n}^{(k)} )_{n \geq 2-k} $ be the $k-$generalized Lucas sequence with certain initial $ k $ terms and each term afterward is the sum of the $ k $ preceding terms. In this paper, we find all repdigits which are products of arbitrary three terms of $k-$generalized Lucas sequences. Thus, we find all non negative integer solutions of Diophantine equation $L_n^{(k)}L_m^{(k)}L_l^{(k)} =a \left( \dfrac{10^{d}-1}{9} \right)$ where $n\geq m \geq l \geq 0$ and $1 \leq a \leq 9.
... To establish the proof of Theorem 1.1, we first find an upper bound for n in terms of k by applying Matveev's result on linear forms in logarithms [8]. When k is small, the theory of continued fractions suffices to lower such bounds and complete the calculations. ...
... With the above notation, Matveev (see [8] or [6, Theorem 9.4]) proved the following result. ...
For an integer $k \geq 2$, let $\{ P_{n}^{(k)} \}_{n}$ be the $k$-generalized Pell sequence which starts with $0, \dots,0,1$($k$ terms) and each term afterwards is the sum of $k$ preceding terms. In this paper, we find all the solutions of the Diophantine equation $P_{n}^{(k)} = N_{m}$ in non-negative integers $(n, k, m)$ with $k \geq 2$, where $\{ N_{m} \}_m$ is the Narayana's cows sequence. Our approach utilizes the lower bounds for linear forms in logarithms of algebraic numbers established by Matveev, along with key insights from the theory of continued fractions.
... In this section, we give necessary lemmas to be used to prove our main theorems in Section 3. The following theorem is made a conclusion from Corollary 2.3 of Matveev [5] (also see Theorem 9.4 in [16]). where A i max f0:16; Dh( i ); j log i jg and B max fjbijg for all i = 1; : : : ; t: ...
... We will now give a lemma that we will use to reduce the upper bound for the subscript k in the equation (5) and which was proved in [8]. It is a version of the lemma given by Dujella and Peth½ o [3]. ...
In this study, we …nd all Perrin numbers (R k) which can be written as a di¤erence of two repdigits or powers of ten. We show that the Perrin numbers which can be written as a di¤erence of two repdigits are R k
... With the above notations, Matveev proved the following result [3]. ...
... log |Λ 3 | > −1.4 · 30 6 · 3 4.5 · 3 2 · (1 + log 3) · (1 + log n) · (4.2 · 10 27 · log 3 ρ · (1 + log n) 2 ) · log α · 3 log ρ > −1.31 · 10 40 · log 4 ρ · (1 + log n) 3 and comparison of this inequality with (19) gives n log α − log 5 < 1.31 · 10 40 · log 4 ρ · (1 + log n) 3 , which satisfies to n < 2.75 · 10 41 · log 4 ρ · log 3 n, with 1 + log n < 2 log n. ...
\noindent Narayana's sequence is a ternary recurrent sequence defined by the recurrence relation $\mathcal{N}_n=\mathcal{N}_{n-1}+\mathcal{N}_{n-3}$ with initial terms $\mathcal{N}_0=0$ and $\mathcal{N}_1=\mathcal{N}_2=\mathcal{N}_3=1$. Let $\rho\geqslant2$ be a positive integer . In this study , it is proved that the $n$ th Narayana number $ \mathcal{N}_n$ which are concatenations of three repdigits in base $\rho$ satisfies $n<5.6\cdot 10^{48}\cdot \log^7\rho$. Moreover , it is shown that the largest Narayana's number which are concatenations of three repdigits in base $\rho$ with $1\leqslant \rho\leqslant 10$ is $58425=\mathcal{N}_{31}=\overline{3332200}_5=\overline{332223}_7 .
... In order to prove our main result, we use a Baker-type lower bound for a non-zero linear forms in logarithms of algebraic numbers a few times. Before presenting a result of Matveev [14] about the general lower bound for linear forms in logarithms, we recall some fundamental notations from algebraic number theory. ...
... With the established notations, Matveev (see [14] or [7, Theorem 9.4]), proved the ensuing result. ...
In this paper,we find all generalized Fibonacci numbers which are Narayana's cows numbers. In our proofs, we use both Baker's theory of nonzero linear forms in logarithms of algebraic numbers and the Baker-Davenport reduction method.
... We use three times Baker-type lower bounds for nonzero linear forms in two or three logarithms of algebraic numbers. There are many such bounds mentioned in the literature like that of Baker and Wüstholz from [2] or Matveev from [10]. Before we can formulate such inequalities we need the notion of height of an algebraic number recalled below. ...
... We write K = Q(α 1 , . . . , α t ) and D for the degree of K. We start with the general form due to Matveev [10]. ...
Let \( \{F_n\}_{n\geq 0} \) be the sequence of Fibonacci numbers and let \(p\) be a prime. For an integer \(c\) we write \(m_{F,p}(c)\) for the number of distinct representations of \(c\) as \(F_k-p^\ell\) with \(k\ge 2\) and \(\ell\ge 0\). We prove that \(m_{F,p}(c)\le 4\).
... Indeed, the degree of the resulting univariate reduction can be so high that a naive use of real root isolation would lead to complexity super-linear in n n/2 d n . So we leverage the special structure of the derivative of our univariate reduction to apply a powerful theorem from diophantine approximation: A refinement of an estimate of Baker and Wustholtz on linear forms in logarithms of algebraic numbers (see, e.g., [4,6,30,77]). ...
... lower bound on the minimum of | (α, b)| over all such α i and b i . Alan Baker won a Fields medal in 1970, due in large part to finding such a bound-over arbitrary number fields-and deriving numerous landmark results in number theory as a consequence [4]: The most recent refinements of his bound [77,80], in the special case of Q, can be coarsely summarized as follows. ...
Suppose A={a1,…,an+2}⊂Zn has cardinality n+2, with all the coordinates of the aj having absolute value at most d, and the aj do not all lie in the same affine hyperplane. Suppose F=(f1,…,fn) is an n×n polynomial system with generic integer coefficients at most H in absolute value, and A the union of the sets of exponent vectors of the fi. We give the first algorithm that, for any fixedn, counts exactly the number of real roots of F in time polynomial in log(dH). We also discuss a number-theoretic hypothesis that would imply a further speed-up to time polynomial in n as well.
... We prove the following results. (1.2) in positive integers (m, n, k) with k ≥ 3 belong to {(1, 1, k), (2, 1, k), (3, 1, k), (4, 2, k), (5, 2, k), (8,3, k)}. ...
... With this notation, we recall Theorem 9.4 of [4], which is a modified version of a result of Matveev [8]. ...
Let $$k\ge 2$$ k ≥ 2 . A generalization of the well-known Pell sequence is the k -Pell sequence. For this sequence, the first k terms are $$0,\ldots ,0,1$$ 0 , … , 0 , 1 and each term afterwards is given by the linear recurrence $$\begin{aligned} P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)}. \end{aligned}$$ P n ( k ) = 2 P n - 1 ( k ) + P n - 2 ( k ) + ⋯ + P n - k ( k ) . In this paper, we extend the previous work (Rihane and Togbé in Ann Math Inform 54:57–71, 2021) and investigate the Padovan and Perrin numbers in the k -Pell sequence.
... These lemmas are very important to prove our theorems. Now, we can give a lemma deduced from Corollary 2.3 of Matveev [8] (also see [4]). The following lemma is used to …nd a large upper bound for k in some equations. ...
In this work, we determine all Pell and Pell-Lucas numbers which are a difference of two repdigits. These are P_k={2; 5; 12; 29; 70} and Q_k={2; 6; 14; 34; 82; 478}, respectively. That is, we solve the Diophantine equations P_k = d1(10^n-1)/9- d2(10^m-1)/9 and Q_ k = d1 (10^n-1)/9 -d2(10^m-1)/9 in non-negative integers (k; m; n; d1; d2). We found that the largest Pell and Pell-Lucas numbers which can be written as difference of two repdig-its are P_6 = 70 = 77-7 and Q_7 = 478 = 555-77. Our proofs based on Baker's theory and reduction method.
... Now we give a theorem which is deduced from Corollary 2.3 of Matveev [8] and provides a large upper bound for the subscript n in the equations (5) and (6)(also see Theorem 9.4 in [3]). ...
In this study, we …nd all Fibonacci numbers F k and Lucas numbers L k which are products of two Jacobsthal-Lucas numbers. More generally, taking k; m; n as nonnegative integers, we proved that F k = jmjn = (2 m + (1) m) (2 n + (1) n) implies that (k; m; n) = (1; 1; 1) ; (2; 1; 1); (3; 0; 1); (5; 1; 2); (9; 0; 4) and L k = jmjn implies that (k; m; n) = (3; 0; 0); (0; 0; 1); (1; 1; 1); (4; 1; 3): As a result of this study, we showed that the largest Fibonacci number and Lucas number which can be written in the form (2 m + (1) m) (2 n + (1) n) are F9 = 34 = 2 17 = (2 0 + (1) 0) (2 4 + (1) 4) and L4 = 7 = 1 7 = (2 1 + (1) 1) (2 3 + (1) 3); respectively. Moreover the largest Fibonacci number and Lucas number which can be written in the form 2 n + (1) n are F5 = 5 = 2 2 + (1) 2 and L4 = 7 = 2 3 + (1) 3 ; respectively. As a result, it is shown that the only Fermat numbers in the Fibonacci sequence are F3 = 3 and F5 = 5 and the only Fermat number in the Lucas sequence is L2 = 3. The proofs depend on lower bounds for linear forms and some tools from Diophantine approximation.
... Then, h(t) 1 3 log 23. The following theorem which is deduced from Corollary 2.3 of Matveev [16] can be found [7]. ...
Let \((P_{k})_{k\ge 0}\) and \((R_{k})_{k\ge 0}\) be the Padovan and Perrin sequences. In this paper, we found that all Padovan numbers, which are concatenations of two Padovan numbers are 12, 21, 37, 49, 265, 465. Moreover, we showed that the only Perrin number, which is concatenations of two Perrin numbers is 22. That is, we solved the Diophantine equations \( P_{k}=10^{d}P_{m}+P_{n}\) and \(R_{k}=10^{d}R_{m}+R_{n}\) in positive integers (k, m, n), where d denotes the number of digits of \(P_{n}\) and \(R_{n}\), respectively. The proofs based on Baker’s theory and we used linear forms in logarithms and reduction method to solve of these Diophantine equations.
... Therefore, we take d ≤ 2. To prove the theorem, we use two effective methods for Diophantine equations. The first tool is linear forms in logarithms of algebraic numbers due to Matveev [9] and, the second one is reduction algorithm due to Dujella and Pethő [6]. To achieve the results, we use the software Maple for all computations. ...
In this paper, we find all repdigits, which are difference of two k−Lucas numbers. The proof depends on lower bounds for linear forms in logarithms of algebraic integers as well as a version of the Baker-Davenport reduction method.
... its absolute logarithmic height. We recall the following result due to Matveev [14]. ...
Let a and b = ka be positive integers with k ∈ {2, 3, 6}, such that ab+4 is a perfect square. In this paper, we study the extensibility of the D(4)-pairs {a, ka}. More precisely, we prove that by considering families of positive integers c depending on a, if {a, b, c, d} is a set of positive integers which has the property that the product of any two of its elements increased by 4 is a perfect square, then d is given by d = a + b + c + 1 2 abc ± (ab + 4)(ac + 4)(bc + 4). As a corollary, we prove that any D(4)-quadruple tht contains the pair {a, ka} is regular.
... The following lemma is derived from Corollary 2.3 of Matveev [14] and allows to …nd a large upper bound for the subscript n. Also see in [4] for this lemma. ...
In this study, we focus on …nding the Pell and Pell-Lucas numbers which are concatenations of three repdigits. We show that this numbers are 169; 408; 985: We use a lemma that provides a large upper bound for the subscript n in the equations and Baker's theory of lower bounds for a nonzero linear form in logarithms of algebraic numbers. In addition, continued fraction expansions of some irrational numbers were calculated to show that some inequalities have no solution.
... With the above notations, Matveev proved the following result [13]. ...
Let P m and E m be the m -th Padovan and Perrin numbers respectively. In this paper, we prove that for a fixed integer δ with δ ≥ 2 there exists a finite Padovan and Perrin numbers that can be representable as products of three repdigits in base δ. Moreover, we explicitly find these numbers for 2 ≤ δ ≤ 10 as an applications.
... From the previous notations, we have the following result due to Bugeaud, Mignotte, and Siksek (see [5,Theorem 9.4]) which is an improved version of Matveev's result [14]. ...
Repdigit in base b is a positive integer that has only one digit in its base b expansion, i.e. a number of the form \(a(b^m-1)/(b-1)\), for some positive integers \(m\ge 1\), \(b\ge 2\) and \(1\le a\le b-1\). In this paper, we investigate all balancing numbers which are expressed as products of three repdigits in base b. As a corollary, we show that the number 35 is the largest balancing number which can be expressible as a product of three repdigits. The proofs use Baker’s theory of lower bounds for nonzero linear forms in logarithms of algebraic numbers and reduction method due to Baker and Davenport, the Dujella-Pethő’s version.
... With the above notations, Matveev proved the following result [12]. ...
Narayana's sequence is a ternary recurrent sequence defined by the recurrence relation $\mathcal{N}_n=\mathcal{N}_{n-1}+\mathcal{N}_{n-3}$ with initial terms $\mathcal{N}_0=0$ and $\mathcal{N}_1=\mathcal{N}_2=\mathcal{N}_3=1$. Let $\rho\geqslant2$ be a positive integer. In this study, it is proved that the $n$th Narayana's number $ \mathcal{N}_n$ which is concatenations of three repdigits in base $\rho$ satisfies $n<5.6\cdot 10^{48}\cdot \log^7\rho$. Moreover, it is shown that the largest Narayana's number which is concatenations of three repdigits in base $\rho$ with $1\leqslant \rho\leqslant 10$ is $58425=\mathcal{N}_{31}=\overline{3332200}_5=\overline{332223}_7 .
... We also need some results from the theory of lower bounds in non-zero linear forms in logarithms of algebraic numbers. We start by recalling Theorem 9.4 of [4], which is a modified version of a result of Matveev [19]. Let L be an algebraic number field of degree d L . ...
In this paper, we look at Diophantine triples with values in three different binary recurrence sequences. These are the Fibonacci and Pell sequences and the sequence of one more of powers of a given prime $p$. The novelty of the article is the appearance of three different sequences, as up to now the analogous problem had been investigated only for one sequence.
... The following theorem can be deduced from Corollary 2.3 of Matveev [14]. ...
In this paper, we study the solutions of the equation $F_n-F_m=p^a$ where $p$ is either $7$ or $13$ and $n>m\geqslant 0$, $a\geqslant 2$. We confirm the conjecture of Erduvan and Keskin by proving that there is no solutions for this Diophantine equation. We will use the lower bounds for linear forms in logarithms (Baker's theory) and a version of the Baker-Davenport reduction method in Diophantine approximation.
... The proof of the above theorem, mainly depends on two effective methods, that is, the linear forms in logarithms of algebraic numbers due to Matveev [21] as well as reduction algorithm due to Dujella and Pethő [13], which is in fact originally introduced by Baker and Davenport in [3]. We give some details of these methods in the next section whereas in the third section, we give the main properties of k−Lucas sequences that we will need later. ...
Let $ k \geq 2 $ and $ ( L_{n}^{(k)} )_{n \geq 2-k} $ be the $k-$generalized Lucas sequence with initial condition $ L_{2-k}^{(k)} = \cdots = L_{-1}^{(k)}=0 ,$ $ L_{0}^{(k,}=2,$ $ L_{1}^{(k)}=1$ and each term afterwards is the sum of the $ k $ preceding terms. A positive integer is an almost repdigit if its digits are all equal except for at most one digit. In this paper, we work on the problem of determining all terms of $k-$generalized Lucas sequences which are almost repdigits. In particular, we find all $k-$generalized Lucas numbers which are powers of $10$ as a special case of almost repdigits.
... Equation (6) and some tricks will allow us to obtain linear forms in three logarithms and then determine lower boundsà la Baker for these linear forms. From the main result of Matveev [7], we deduce the following lemma. Lemma 1. ...
Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_{n+2}=F_{n+1}+F_n$, for $n\geq 0$, where $F_0=0$ and $F_1=1$. There are several interesting identities involving this sequence such as $F_n^2+F_{n+1}^2=F_{2n+1}$, for all $n\geq 0$. In a very recent paper, Chaves, Marques and Togb\' e proved that if if $(G_m)_m$ is a linear recurrence sequence (under weak assumptions) and $G_n^s+\cdots +G_{n+k}^s\in (G_m)_m$, for infinitely many positive integers $n$, then $s$ is bounded by an effectively computable constant depending only on $k$ and the parameters of $G_m$. In this paper, we will generalize this result, proving, in particular, that if $(G_m)_m, (H_m)_m$ are linear recurrence sequences (under weak assumptions) and $ \e_0R(G_n)+\e_1R(G_{n+1})+\cdots +\e_{k-1}R(G_{n+k-1})+R(G_{n+k})$, for infinitely many positive integers $n$, then $s$ is bounded by an effectively computable constant depending only on upper and lower bounds of the $\e_i$ and the parameters of $G_m$ (but surprisingly not on $k$).
... The following theorem of Matveev (see [7] or [2, Theorem 9.4]) provides a large upper bound for the subscript n in (1.2). ...
In this note we solve the Diophantine equation Nn = x a ±x b +1, where Nn denotes the n-th Narayana number, a, b are nonnegative integers with 0 ≤ b < a and 2 ≤ x ≤ 30.
... There are several different estimates of Bakertype lower bounds for linear forms in logarithms. In this study, we use the most common Baker type method due to Matveev [13] or [3,Theorem 9.4]. Thus, our main results are as follows. ...
Recall that repdigit in base $g$ is a positive integer that has only one digit in its base $g$ expansion, i.e. a number of the form $a(g^m-1)/(g-1)$, for some positive integers $m\geq 1$, $g\geq 2$ and $1\leq a\leq g-1$. In the present study we investigate all Fibonacci or Lucas numbers which are expressed as products of three repdigits in base $g$. As illustration, we consider the case $g=10$ where we show that the numbers 144 and 18 are the largest Fibonacci and Lucas numbers which can be expressible as products of three repdigits respectively. All this can be done using linear forms in logarithms of algebraic numbers.
... The following theorem is a modified version of a result of Matveev (see [8] or [4, Theorem 9.4]) which provides a large upper bound for n 1 in (2). ...
In this study we find all solutions of the Diophantine equation $B_{n_{1}}+B_{n_{2}}=2^{a_{1}}+2^{a_{2}}+2^{a_{3}}$ in positive integer variables $(n_{1},n_{2},a_{1},a_{2},a_{3}),$ where $B_{n}$ denotes the $n$-th balancing number.
... The following is an implication of Matveev's [4] monumental result on bounding linear forms in logarithms given by Bugeaud, Mignotte, and Siksek [2] to solve exponential Diophantine equations. We will use their version here for linear recurrences. ...
In this paper, we consider the Diophantine equation $\lambda_1U_{n_1}+\ldots+\lambda_kU_{n_k}=wp_1^{z_1} \cdots p_s^{z_s},$ where $\{U_n\}_{n\geq 0}$ is a fixed non-degenerate linear recurrence sequence of order greater than or equal to 2; $w$ is a fixed non-zero integer; $p_1,\dots,p_s$ are fixed, distinct prime numbers; $\lambda_1,\dots,\lambda_k$ are strictly positive integers; and $n_1,\dots,n_k,z_1,\dots,z_s$ are non-negative integer unknowns. We prove the existence of an effectively computable upper-bound on the solutions $(n_1,\dots,n_k,z_1,\dots,z_s)$. In our proof, we use lower bounds for linear forms in logarithms, extending the work of Pink and Ziegler (2016), Mazumdar and Rout (2019), Meher and Rout (2017), and Ziegler (2019).
... The following theorem is a modified version of a result of Matveev (see [8] or [4,Theorem 9.4]) which provides a large upper bound for n 1 in (1.2). ...
In this study we find all solutions of the Diophantine equation $B_{n_{1}}+B_{n_{2}}=2^{a_{1}}+2^{a_{2}}+2^{a_{3}}$ in positive integer variables $(n_{1},n_{2},a_{1},a_{2},a_{3}),$ where $B_{n}$ denotes the $n$-th balancing number.
... With these notations, Matveev (see [10] or [5,Theorem 9.4]) proved the following result. ...
In this study, we find all Narayana numbers which are expressible as sums of two base b repdigits. The proof of the main result uses lower bounds for linear forms in logarithms of algebraic numbers and a version of the Baker–Davenport reduction method.
... With the above notations, Matveev (see [12] or [6,Theorem 9.4]) proved the following result. ...
For any integer $k \geq 2$, let $\{Q_{n}^{(k)} \}_{n \geq -(k-2)}$ denote the $k$-generalized Pell-Lucas sequence which starts with $0, \dots ,2,2$($k$ terms) where each next term is the sum of the $k$ preceding terms. In this paper, we find all the $k$-generalized Pell-Lucas numbers that are the product of two repdigits. This generalizes a result of Erduvan and Keskin \cite{Erduvan1} regarding repdigits of Pell-Lucas numbers.
... The proof of the following result can be found in [9]. Theorem 2.1. ...
We completely solve the two Diophantine equation B k = J n + J m and C k = J n + J m where {B k } k≥0 , {C k } k≥0 and {J k } k≥0 are the sequences of balancing , Lucas-balancing and Jacobsthal numbers, respectively. We use Matveev's theorem on linear forms in logarithms and a related reduction procedure of Dujella and Pethö .
... The proof of the following result can be found in [7]. ...
We solve the two Diophantine equations $P_k=J_n+J_m$ and $Q_k=J_n+J_m$ where $\left\lbrace P_{k}\right\rbrace_{k\geq0}$, $\left\lbrace Q_{k}\right\rbrace_{k\geq0}$ and $\left\lbrace J_{k}\right\rbrace_{k\geq0}$ are the sequences of Pell numbers, Pell-Lucas numbers and Jacobsthal numbers, respectively. The main tool is the theory of linear forms in logarithms.
... The first tool we need is the following result due to Matveev [14]. Here we use the version of Bugeaud, Mignotte and Siksek [3, Theorem 9.4]. ...
Let ≥ 2 be an integer. In this paper we study the base repdigits that can be expressed as sums or products of Fi-bonacci and Tribonacci numbers. As a corollary, it is shown that the numbers 1 and 7 are the only Mersenne numbers which can be expressed respectively as product and sum of Fibonacci and Tribonacci numbers. This can be done using linear forms in logarithms of algebraic numbers and the reduction method due to Dujella and Pethő.
... The main ingredient of our proofs is an effective lower bound for linear forms in logarithms of algebraic numbers. We use here a theorem of Matveev (2000). Let K be a number field of degree D over Q, let η 1 , η 2 , . . . ...
The integer sequence defined by P_{n+3}=P_{n+1}+P_{n} with initial conditions P_{0}=1 and P_{1}=P_{2}=0 is known as the Padovan sequence {P_n}_{n∈Z}. A recurrence sequence {u_n}_{n∈Z} is said to be of Padovan-type if it satisfies the same recurrence relation as the Padovan sequence but with arbitrary initial values u_{0}, u_{1}, u_{2} not all zero. The most famous Padovan-type sequence is the Perrin sequence given by u_{0}=3, u_{1}=0 and u_{2}=2. We show that every Padovan-type sequence has at most 2 zeros, except for nonzero multiples of shifts of the Padovan sequence which has 0-multiplicity 5. We also show that {P_n}_{n∈Z} has total multiplicity 62, i.e., there are 62 pairs (m,n)∈Z^2 with m<n for which P_{m}=P_{n}. As a consequence, we found that {P_n}_{n∈Z} has multiplicity 9, being 1 the most repeated Padovan number. Finally, we prove that the Perrin sequence has exactly 1 zero, total multiplicity 23 and multiplicity 4.
... This leads to a contradiction. □ Now, we use another theorem for the lower bounds of linear forms in logarithms from Matveev [12], which is quoted below. ...
Let k ≥ 1 be an integer and let P_k and Q_k be the k-th Pell number and k-th Pell-Lucas number, respectively. In this paper, we prove that if d is a positive integer such that {P_{2k} , P_{2k+2}, 2P_{2k+2} , d} is a Diophantine quadruple, then d = P_{2k+1} Q_{2k+1} Q_{2k+2}. We deduce that the pair {P_{2k} , 2P_{2k+2} } cannot be extended to an irregular Diophantine quadruple.
In this study, it is proved that the only Fibonacci numbers which are con-catenations of three repdigits are 144, 233, 377, 610, 987, 17711.
In this study, it is proved that the only Fibonacci numbers which are con-catenations of three repdigits are 144, 233, 377, 610, 987, 17711.
Here, we find all positive integer solutions of the Diophantine equation in the title, where $(\mathcal{U}_n)_{n\geqslant 0}$ is the generalized Lucas sequence $\mathcal{U}_0=0, \ \mathcal{U}_1=1$ and $\mathcal{U}_{n+1}=r \mathcal{U}_n +s \mathcal{U}_{n-1}$ with $r$ and $s$ integers such that $\Delta = r^2 +4 s >0$.
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–Pethö version of the Baker–Davenport reduction method in diophantine approximation. This paper continues and extends the previous work of Chern and Cui.
Let S={p1,…,ps} be a finite, non-empty set of distinct prime numbers and (Un)n≥0 be a linear recurrence sequence of integers of order at least 2. For any positive integer k, and w=(wk,…,w1)∈Zk,w1,…,wk≠0 we define (Uj(k,w))j≥1 an increasing sequence composed of integers of the form |wkUnk+⋯+w1Un1|, nk>⋯>n1. Under certain assumptions, we prove that for any ε>0, there exists an integer n0 such that [Uj(k,w)]S<(Uj(k,w))ε,forj>n0, where [m]S denotes the S-part of the positive integer m. On further assumptions on (Un)n≥0, we also compute an effective bound for [Uj(k,w)]S of the form (Uj(k,w))1-c, where c is a positive constant depending only on r, a1 , . . ., ar, U0, . . ., Ur-1 , w1, . . ., wk and S.
In this paper, we improve a result of Fujita and Le concerning the Diophantine equation x2+(2c-1)y=cZ
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.
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.
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.
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
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.