In this paper, it is shown that if F(x, y) is an irreducible binary form with integral coefficients and degree n >= 3, then provided that the absolute value of the discriminant of F is large enough, the equation F(...
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In this paper, it is shown that if F(x, y) is an irreducible binary form with integral coefficients and degree n >= 3, then provided that the absolute value of the discriminant of F is large enough, the equation F(x, y) = +/- 1 has at most 11n - 2 solutions in integers x and y. We will also establish some sharper bounds when more restrictions are assumed. These upper bounds are derived by combining methods from classical analysis and geometry of numbers. The theory of linear forms in logarithms plays an essential role in studying the geometry of our Diophantine equations.
Let (un)n >= 0 be a non-degenerate binary recurrence sequence with positive discriminant and p be a fixed prime number. In this paper, we are interested in finding a finiteness result for the solutions of the Dioph...
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Let (un)n >= 0 be a non-degenerate binary recurrence sequence with positive discriminant and p be a fixed prime number. In this paper, we are interested in finding a finiteness result for the solutions of the Diophantine equation u(n1) + u(n2) + . . . + u(nt) = p(z) with n(1) > n(2) >... > n(t) >= 0. Moreover, we explicitly find all the powers of three which are sums of three balancing numbers using lower bounds for linear forms in logarithms. Further, we use a variant of the Baker-Davenport reduction method in Diophantine approximation due to Dujella and Petho.
The Euler-Lehmer constants gamma(a, q) are defined as the limits lim(x ->infinity) ( Sigma(n = 2, is an algebraic number. The methods used to prove this theorem can also be applied to study the following question o...
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The Euler-Lehmer constants gamma(a, q) are defined as the limits lim(x ->infinity) ( Sigma(n <= x) 1/n - logx/q) n equivalent to a (mod q) We show that at most one number in the infinite list gamma(a,q), 1 <= a < q, q >= 2, is an algebraic number. The methods used to prove this theorem can also be applied to study the following question of Erdos. If f : Z/qZ -> Q is such that f(a) = +/- 1 and f(q) = 0, then Erdos conjectured that Sigma(infinity)(n=1) f(n)/n not equal 0. If q equivalent to 3 (mod 4), we show that the Erdos conjecture is true. (C) 2010 Elsevier Inc. All rights reserved.
In this study, we find all Fibonacci and Lucas numbers which can be expressible as a product of two repdigits in the base b. It is shown that the largest Fibonacci and Lucas numbers which can be expressible as a produ...
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In this study, we find all Fibonacci and Lucas numbers which can be expressible as a product of two repdigits in the base b. It is shown that the largest Fibonacci and Lucas numbers which can be expressible as a product of two repdigits are F-12 = 144 and L-15 = 1364, respectively. Also, we have the presentation F-12 = 144 = 6 x (3 + 3. 7) = (6)(7) x (33)(7) = 4 x (4 + 4.8) = (4)(8) x (44)(8) and L-15 = 1364 x (22222)(4) = 2 x (2 + 2.4 + 2. 4(2) + 2.4(3) + 2.4(4)).
Let k, m and n be integers. In this paper, for a fixed integer mu not equal 0, we show that the family of Thue equation x(4) - kmnn(3)y + (km(2) - km(2) + 2)x(2)y(2) + kmnxy(3) + y(4) = mu is reducible by Tzanakis'...
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Let k, m and n be integers. In this paper, for a fixed integer mu not equal 0, we show that the family of Thue equation x(4) - kmnn(3)y + (km(2) - km(2) + 2)x(2)y(2) + kmnxy(3) + y(4) = mu is reducible by Tzanakis's method into a system of pellian equations kV(2) - (km(2) + 4)U-2 = -4 mu, kZ(2) - (kn(2) - 4)U-2 = 4 mu, with any triple of integers (k, m, n) such that k > 0, vertical bar n vertical bar >= 2, vertical bar m vertical bar >= 2. We consider this system for any even integer k not equal 2 square, mu = 1 and we prove that for all integers vertical bar n vertical bar >= 2 and vertical bar m vertical bar >= 2 that are sufficiently large and have sufficiently large common divisor this system has only the trivial solutions (U, V, Z,) = ( 1, m, n). We also show that if k not equal 2 square is even, then the system has in general at most 8 solutions in positive integers. (C) 2019 Elsevier Inc. All rights reserved.
For k a parts per thousand yen 2, the k-generalized Fibonacci sequence (F (n) ((k)) ) (n) is defined by the initial values 0,0,aEuro broken vertical bar, 0, 1 (k terms) so that each term afterward is the sum of the k ...
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For k a parts per thousand yen 2, the k-generalized Fibonacci sequence (F (n) ((k)) ) (n) is defined by the initial values 0,0,aEuro broken vertical bar, 0, 1 (k terms) so that each term afterward is the sum of the k preceding terms. In this paper, we prove that the only solution of the Diophantine equation F (m) ((k)) = k (t) = k (t) with t > 1 and m > k+ 1 a parts per thousand yen 4 is F (9) ((3)) = 3(4).
A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by unity is a perfect square. Any Diophantine triple is conjectured to be uniquely extended to a Diophan...
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A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by unity is a perfect square. Any Diophantine triple is conjectured to be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element in the triple. A previous work of the second and third authors revealed that the number of such extensions for a fixed Diophantine triple is at most 11. In this paper, we show that the number is at most eight.
We will give upper bounds upon the number of integral solutions to binary quartic Thue equations. We will also study the geometric properties of a specific family of binary quartic Thue equations to establish sharper ...
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We will give upper bounds upon the number of integral solutions to binary quartic Thue equations. We will also study the geometric properties of a specific family of binary quartic Thue equations to establish sharper upper bounds. (C) 2009 Elsevier Inc. All rights reserved.
Under certain assumptions, it is shown that eq. (2) has only finitely many solutions in integersx≥0,y≥0,k≥2,l≥0. In particular, it is proved that (2) witha=b=1, l=k implies thatx=7,y=0,k=3.
Under certain assumptions, it is shown that eq. (2) has only finitely many solutions in integersx≥0,y≥0,k≥2,l≥0. In particular, it is proved that (2) witha=b=1, l=k implies thatx=7,y=0,k=3.
For an integer , let be the -Fibonacci sequence which starts with ( terms) and each term afterwards is the sum of the preceding terms. In this paper, we find all repdigits (i.e., numbers with only one distinct digit i...
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For an integer , let be the -Fibonacci sequence which starts with ( terms) and each term afterwards is the sum of the preceding terms. In this paper, we find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion) which are sums of two -Fibonacci numbers. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a version of the Baker-Davenport reduction method. This paper is an extended work related to our previous work (Bravo and Luca Publ Math Debr 82:623-639, 2013).
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