In this paper, we consider the Diophantine equation lambda U-1(n1) + center dot center dot center dot + lambda U-k(nk) = wp(1)(z1) ... p(s)(zs), where {U-n}(n >= 0) is a fixed non-degenerate linear recurrence seque...
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In this paper, we consider the Diophantine equation lambda U-1(n1) + center dot center dot center dot + lambda U-k(nk) = wp(1)(z1) ... p(s)(zs), where {U-n}(n >= 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), ... , p(s) are fixed, distinct prime numbers;lambda(1), ... , lambda(k) are strictly positive integers;and n(1), ... , n(k), z(1), ... , z(s) are non-negative integer unknowns. We prove the existence of an effectively computable upper-bound on the solutions (n(1), ... , n(k), z(1), ... , z(s)). In our proof, we use lower bounds for linear forms in logarithms, extending the work of Pink and Ziegler (Monatshefte Math 185(1):103-131, 2018), Mazumdar and Rout (Monatshefte Math 189(4):695-714, 2019), Meher and Rout (Lith Math J 57(4):506-520, 2017), and Ziegler (Acta Arith 190:139-169, 2019).
For k >= 2, the sequence (F-n((k)))(n >=-(k-2)) of k-generalized Fibonacci numbers is defined by the initial values 0, ..., 0, 1 = F-1((k)) and such that each term afterwards is the sum of the k preceding ones. ...
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For k >= 2, the sequence (F-n((k)))(n >=-(k-2)) of k-generalized Fibonacci numbers is defined by the initial values 0, ..., 0, 1 = F-1((k)) and such that each term afterwards is the sum of the k preceding ones. There are many recent results about the Diophantine equation (F-n((k)))(s) + (F-n+1((k)))(s) = F-m((l)), most of them dealing with the case k = l. In 2018, Bednarik et al. solved the equation for k <= l, but with s = 2. The aim of this paper is to continue this line of investigation by solving this equation for all s >= 2, but with (k, l) = (3, 2).
The k-generalized Pell sequence P-(k) := (P-n((k)))(n >= -(k-2)) is the linear recurrence sequence of order k, whose first k terms are 0, ..., 0, 1 and satisfies the relation P-n((k)) = 2P(n-1)((k)) + 2P(n-2)((k)) ...
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The k-generalized Pell sequence P-(k) := (P-n((k)))(n >= -(k-2)) is the linear recurrence sequence of order k, whose first k terms are 0, ..., 0, 1 and satisfies the relation P-n((k)) = 2P(n-1)((k)) + 2P(n-2)((k)) + ... + P-n-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 P-n((k)) - b(m) for b is an element of [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).
In this paper, we show that 204 and 1189 are the only balancing numbers which are concatenation of three repdigits and that 3363 is the only Lucas-balancing number of this form.
In this paper, we show that 204 and 1189 are the only balancing numbers which are concatenation of three repdigits and that 3363 is the only Lucas-balancing number of this form.
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 ...
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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.
Let (F-n)(n >= 0) be a Fibonacci sequence. A non-negative integer whose digits are all equal is called a repdigit and any non-zero repdigit is of the form a (10(d) -1/9) where 1 <= a <= 9 and 1 <= d. In th...
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Let (F-n)(n >= 0) be a Fibonacci sequence. A non-negative integer whose digits are all equal is called a repdigit and any non-zero repdigit is of the form a (10(d) -1/9) where 1 <= a <= 9 and 1 <= d. In this paper, we search all repdigits that can be written as products of three Fibonacci numbers. As a mathematical expression, we find all non-negative integer solutions (n, m, l, a, d) of the Diophantine equation FnFmFl = a (10(d) -1/9) , 1 <= l <= m <= n and 1 <= a <= 9..
For a number field K, let zeta(K) (s) be the Dedekind zeta function associated to K. In this paper, we study non-vanishing and transcendence of zeta(K) as well as its derivative CK at s = 1/2. En route, we strengthen ...
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For a number field K, let zeta(K) (s) be the Dedekind zeta function associated to K. In this paper, we study non-vanishing and transcendence of zeta(K) as well as its derivative CK at s = 1/2. En route, we strengthen a result proved by Ram Murty and Tanabe [On the nature of e(gamma) and non-vanishing of L-series at s = 1/2, J. Number Theory 161 (2016) 444 456].
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 fo...
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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-Petho version of the Baker-Davenport reduction method in diophantine approximation. This paper continues and extends the previous work of Chern and Cui.
Recall that a 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= 1, g = 2 and 1 = a = g-1. In the pre...
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Recall that a 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= 1, g = 2 and 1 = a = 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 is done using linear forms in logarithms of algebraic numbers.
Let n not equal 0 be an integer. A set of m distinct positive integers {a(1), a(2), ..., a(m)} is called a D(n)-m-tuple if a(i)a(j) +n is a perfect square for all 1 1, then d = 1. The proof relies not only on standar...
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Let n not equal 0 be an integer. A set of m distinct positive integers {a(1), a(2), ..., a(m)} is called a D(n)-m-tuple if a(i)a(j) +n is a perfect square for all 1 <= i < j <= m. Let k be a positive integer. In this paper, we prove that if {k, k + 1, c, d) is a D(-k)-quadruple with c > 1, then d = 1. The proof relies not only on standard methods in this field (Baker's linear forms in logarithms and the hypergeometric method), but also on some less typical elementary arguments dealing with recurrences, as well as a relatively new method for the determination of integral points on hyperelliptic curves.
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