In this paperwe prove that if {a, b, c} is a Diophantine triple with a < b < c, then{a+1, b, c} cannot be a Diophantine triple. Moreover, we show that if {a1, b, c} and {a(2), b, c} are Diophantine triples with ...
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In this paperwe prove that if {a, b, c} is a Diophantine triple with a < b < c, then{a+1, b, c} cannot be a Diophantine triple. Moreover, we show that if {a1, b, c} and {a(2), b, c} are Diophantine triples with a1 < a(2) < b < c < 16b(3), then{a(1), a(2), b, c} is a Diophantine quadruple. In view of these results, we conjecture that if {a1, b, c} and {a(2), b, c} are Diophantine triples with a(1) < a(2) < b < c, then {a(1), a(2), b, c} is a Diophantine quadruple.
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 w...
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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.
Let (L-n) be the sequence of Lucas numbers defined byL(0)= 2, L-1= 1, andL(n)=Ln-1+L(n-2)forn >= 2. Let 0 <= m <= nandb= 2,3,4,5,6,7,8,*** this study, we show that ifL(m)L(n)is a repdigit in the baseband has ...
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Let (L-n) be the sequence of Lucas numbers defined byL(0)= 2, L-1= 1, andL(n)=Ln-1+L(n-2)forn >= 2. Let 0 <= m <= nandb= 2,3,4,5,6,7,8,*** this study, we show that ifL(m)L(n)is a repdigit in the baseband has at least two digits, then LmLn is an element of {3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 28, 36, 54, 121, 228}. Furthermore, it is shown that ifL(n)is a repdigit in the baseband has at least two digits, then (n, b) = (2,2),(3,3),(4,6),(4,2),(6,5),(6,8). Namely, L-2= 3 = (11)(2),L-3= 4 = (11)(3),L-4= 7 = (11)(6) and L-4= 7 = (111)(2),L-6= 18 = (33)(5),L-6= 18 = (22)(8).
In this study, it is shown that the only Lucas numbers which are concatenations of three repdigits are 123, 199, 322, 521, 843, 2207, 5778. The proof depends on lower bounds for linearforms and some tools from Diopha...
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In this study, it is shown that the only Lucas numbers which are concatenations of three repdigits are 123, 199, 322, 521, 843, 2207, 5778. The proof depends on lower bounds for linearforms and some tools from Diophantine approximation.
The goal of this article is to associate a p-adic analytic function to the Euler constants gamma(p)(a, F), study the properties of these functions in the neighborhood of s = 1 and introduce a p-adic analogue of the in...
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The goal of this article is to associate a p-adic analytic function to the Euler constants gamma(p)(a, F), study the properties of these functions in the neighborhood of s = 1 and introduce a p-adic analogue of the infinite sum Sigma(n >= 1) f(n)/n for an algebraic valued, periodic function f. After this, we prove the theorem of Baker, Birch and Wirsing in this setup and discuss irrationality results associated to p-adic Euler constants generalising the earlier known results in this direction. Finally, we define and prove certain properties of p-adic Euler-Briggs constants analogous to the ones proved by Gun, Saha and Sinha.
In this paper, we study the solutions to the titular Diophantine equation in integers n >= m >= 0, y >= 2 and a >= 2. We show that there are only finitely many of them for a fixed y, and we provide a bound...
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In this paper, we study the solutions to the titular Diophantine equation in integers n >= m >= 0, y >= 2 and a >= 2. We show that there are only finitely many of them for a fixed y, and we provide a bound on the largest such solution. As an application, we find all the solutions when y is an element of [2, 1000]. We also show that the abc-conjecture implies that there are only finitely many integer solutions (n, m, y, a) with min{y, a} >= 2. (c) 2020 Elsevier Inc. All rights reserved.
Let S = {p1,..., pt} be a fixed finite set of prime numbers listed in increasing order. In this paper, we prove that the Diophantine equation ( F (k) n)s = pa1 1 + ... + pat t, in integer unknowns n = 1, s = 1, k = 2 ...
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Let S = {p1,..., pt} be a fixed finite set of prime numbers listed in increasing order. In this paper, we prove that the Diophantine equation ( F (k) n)s = pa1 1 + ... + pat t, in integer unknowns n = 1, s = 1, k = 2 and ai = 0 for i = 1,..., t such that max {ai : 1 = i = t} = at has only finitely many effectively computable solutions. Here, F (k) n is the nth k-generalized Fibonacci number. We compute all these solutions when S = {2, 3, 5}. This paper extends the main results of [15] where the particular case k = 2 was treated.
Let (F-n) be the sequence of Fibonacci numbers defined by F-0 = 0, F-1 = 1, and F-n = Fn-1 + Fn-2 for n >= 2. Let 2 <= m <= n and b = 2, 3, 4, 5, 6, 7, 8, 9. In this study, we show that if FmFn is a repdigit ...
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Let (F-n) be the sequence of Fibonacci numbers defined by F-0 = 0, F-1 = 1, and F-n = Fn-1 + Fn-2 for n >= 2. Let 2 <= m <= n and b = 2, 3, 4, 5, 6, 7, 8, 9. In this study, we show that if FmFn is a repdigit in base b and has at least two digits, then FmFn is an element of {3, 4, 5, 6, 8, 9, 10, 13, 15, 16, 21, 24, 26, 40, 42, 63, 170, 273}. Furthermore, it is shown that if F-n is a repdigit in base b and has at least two digits, then (n, b) = (7, 3), (8, 4), (8, 6), (4, 2), (5, 4), (6, 3), (6, 7).
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