In this paper, we study sets of positive integers with the property that the product of any two elements in the set increased by the unity is a square. It is shown that if the two smallest elements have the form KA(2)...
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In this paper, we study sets of positive integers with the property that the product of any two elements in the set increased by the unity is a square. It is shown that if the two smallest elements have the form KA(2), 4KA(4) +/- 4A for some positive integers A and K, and the third one is chosen canonically, then any such set consisting of three elements can be contained in a unique such set with four elements.
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...
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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 T-2 + 2T, 4T(4) + 8T(3) - 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 KA(2), 4KA(4) +/- 4A for some positive integers A and K is an element of{1,2,3,4}. (C) 2019 Elsevier Inc. All rights reserved.
Recently, Gun, Saha and Sinha had introduced the notion of generalised Euler Briggs constant gamma(Omega, a, q) for a finite set of primes Omega. In a subsequent work, Gun, Murty and Saha introduced the following Q-ve...
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Recently, Gun, Saha and Sinha had introduced the notion of generalised Euler Briggs constant gamma(Omega, a, q) for a finite set of primes Omega. In a subsequent work, Gun, Murty and Saha introduced the following Q-vector space V-Q,V-N = Q , and showed that dim(Q) V-Q,V-N >>(Omega) N. In this note, we improve the lower bound, namely dim(Q) V-Q,V-N >>(Omega) N-2/log N.
It is known that the Fourier-Stieltjes coe fficients of a nonatomic coin-tossing measure may not vanish at infinity. However, we show that they could vanish at infinity along some integer subsequences, including the s...
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It is known that the Fourier-Stieltjes coe fficients of a nonatomic coin-tossing measure may not vanish at infinity. However, we show that they could vanish at infinity along some integer subsequences, including the sequence {b(n)}(n >= 1) where b is multiplicatively independent of 2 and the sequence given by the multiplicative semigroup generated by 3 and 5. The proof is based on elementary combinatorics and lower-bound estimates for linear forms in logarithms from transcendental number theory.
In this paper, we give an algorithm which finds, for an integer base b >= 2, all squarefree integers d >= 2 such that sequence of X-components {X-n}(n >= 1) of the Pell equation X-2-dY(2) = +/- 1 has two memb...
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In this paper, we give an algorithm which finds, for an integer base b >= 2, all squarefree integers d >= 2 such that sequence of X-components {X-n}(n >= 1) of the Pell equation X-2-dY(2) = +/- 1 has two members which are baseb-repdigits. We implement this algorithm and find all the solutions to this problem for all bases b is an element of[2, 100].
Let (Fn)(n >= 0) be the Fibonacci sequence given by Fn+2 = Fn+1+F-n for n >= 0 where F0=0 F-1 = 1. In this paper, we explicitly find all solutions of the title Diophantine equation using lower bounds for linear ...
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Let (Fn)(n >= 0) be the Fibonacci sequence given by Fn+2 = Fn+1+F-n for n >= 0 where F0=0 F-1 = 1. 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 Petho
In this paper, we find all Padovan and Perrin numbers which can be expressible as a products of two repdigits in the base b with 2 <= b <= 10. It is shown that the largest Padovan and Perrin numbers which can be...
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In this paper, we find all Padovan and Perrin numbers which can be expressible as a products of two repdigits in the base b with 2 <= b <= 10. It is shown that the largest Padovan and Perrin numbers which can be expressible as a products of two repdigits are P-25 = 616 and T-22 = 486, respectively. The proofs use lower bounds for linearforms in three logarithms of algebraic numbers and some tools from Diophantine approximation.
Consider the sequence {Fn} n=0 of Fibonacci numbers defined by F0 = 0, F1 = 1, and Fn+ 2 = Fn+ 1 + Fn for all n = 0. In this paper, we find all integers c having at least two representations as a difference between a ...
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Consider the sequence {Fn} n=0 of Fibonacci numbers defined by F0 = 0, F1 = 1, and Fn+ 2 = Fn+ 1 + Fn for all n = 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.
For an integer d >= 2 which is not a square, we show that there is at most one value of the positive integer X participating in the Pell equation X-2 - dY(2) = +/- 1 which is a product of two Fibonacci numbers, wit...
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For an integer d >= 2 which is not a square, we show that there is at most one value of the positive integer X participating in the Pell equation X-2 - dY(2) = +/- 1 which is a product of two Fibonacci numbers, with a few exceptions that we completely characterize. (C) 2019 Elsevier Inc. All rights reserved.
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