This paper is concerned with the existence of consecutive pairs and consecutive triples of multiplicatively dependent integers. A theorem by LeVeque on Pillai's equation implies that the only consecutive pairs of ...
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This paper is concerned with the existence of consecutive pairs and consecutive triples of multiplicatively dependent integers. A theorem by LeVeque on Pillai's equation implies that the only consecutive pairs of multiplicatively dependent integers larger than 1 are (2, 8) and (3, 9). For triples, we prove the following theorem: If a is not an element of & nbsp;{2, 8} is a fixed integer larger than 1, then there are only finitely many triples (a, b, c) of pairwise distinct integers larger than 1 such that (a, b, c), (a +1, b +1, c + 1) and (a +2, b +2, c +2) are each multiplicatively dependent. Moreover, these triples can be determined effectively. (C)& nbsp;2021 The Authors. Published by Elsevier Inc.
The arithmetic nature of the Euler's constant gamma is one of the biggest unsolved problems in number theory from almost three centuries. In an attempt to give a partial answer to the arithmetic nature of gamma, M...
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The arithmetic nature of the Euler's constant gamma is one of the biggest unsolved problems in number theory from almost three centuries. In an attempt to give a partial answer to the arithmetic nature of gamma, Murty and Saradha made a conjecture on linear independence of digamma values. In particular, they conjectured that for any positive integer q > 1 and a field K over which the q-th cyclotomic polynomial is irreducible, the digamma values namely psi (a/q) where 1 ( )<= a <= q with (a, q) = 1 are linearly independent over K. Further, they established a connection between the arithmetic nature of the Euler's constant gamma to the above conjecture. In this article, we first prove that the conjecture is true with at most one exceptional q. Later on we also make some remarks on the linear independence of these digamma values with the arithmetic nature of the Euler's constant gamma.
Let (F-n)(n >= 0) and (L-n)(n >= 0) be the Fibonacci and Lucas sequences. In this paper we determine all Fibonacci and Lucas numbers which are concatenations of two terms of the other sequence. This problem is i...
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Let (F-n)(n >= 0) and (L-n)(n >= 0) be the Fibonacci and Lucas sequences. In this paper we determine all Fibonacci and Lucas numbers which are concatenations of two terms of the other sequence. This problem is identical to solve the Diophantine equations F-n = 10(d) L-m + L-k and L-n = 10(d) F-m + F-k in non-negative integers (n, m, k), where d denotes the number of digits of L-k and F-k, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.
We prove that the trace of the Hecke operator T-2 acting on the vector space of cusp forms of level one takes no repeated values, except for 0, which only occurs when the space is trivial.(c) 2022 Royal Dutch Mathemat...
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We prove that the trace of the Hecke operator T-2 acting on the vector space of cusp forms of level one takes no repeated values, except for 0, which only occurs when the space is trivial.(c) 2022 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
Let (P-n)(n >= 0) be the Pell sequence defined by P-0 = 0, P-1 = 1 and Pn+2 2P(n+1) + P-n for n >= 0 and M-k + 2(k) - 1 be the k-th Mersenne number for k >= 1: We show that the Diophantine equation PnPm + M-k...
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Let (P-n)(n >= 0) be the Pell sequence defined by P-0 = 0, P-1 = 1 and Pn+2 2P(n+1) + P-n for n >= 0 and M-k + 2(k) - 1 be the k-th Mersenne number for k >= 1: We show that the Diophantine equation PnPm + M-k with m <= n has only the unique positive integer solution (n, m, k) + (1;1;1).
Let (F-n)(n >= 0) and (P-n)(n >= 0) be the Fibonacci and the Padovan sequences given by the initial conditions F-0 = 0, F-1 = 1, P-0 = 0, P-1 = P-2 = 1 and the recurrence formulas Fn+2 = Fn+1 + F-n, Pn+3 = Pn+1 ...
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Let (F-n)(n >= 0) and (P-n)(n >= 0) be the Fibonacci and the Padovan sequences given by the initial conditions F-0 = 0, F-1 = 1, P-0 = 0, P-1 = P-2 = 1 and the recurrence formulas Fn+2 = Fn+1 + F-n, Pn+3 = Pn+1 + Pn for all n >= 0, respectively. In this note we study and completely solve the Diophantine equation P-n + P-m = F-l in non-negative integers (n, m, l).
Let (P-n)(n >= 0) be the Padovan sequence given by P-0 = 0, P-1 = P-2 = 1 and the recurrence formula Pn+3 = Pn+1 + P-n for all n >= 0. In this note, we completely solve the Diophantine equation P-m = P-n(x) + P-...
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Let (P-n)(n >= 0) be the Padovan sequence given by P-0 = 0, P-1 = P-2 = 1 and the recurrence formula Pn+3 = Pn+1 + P-n for all n >= 0. In this note, we completely solve the Diophantine equation P-m = P-n(x) + P-n+1(x) in non-negative integers (m, n, x).
In this paper,we find all repdigits expressible as difference of two Fibonacci numbers in base b for 2≤b≤*** largest repdigits in base b,which can be written as difference of two Fibonacci numbers are F9-F4=34-3=31=...
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In this paper,we find all repdigits expressible as difference of two Fibonacci numbers in base b for 2≤b≤*** largest repdigits in base b,which can be written as difference of two Fibonacci numbers are F9-F4=34-3=31=(11111)2,F14-F7=377-13=364=(111111)3,F14-F7=377-13=364=(222)4,F9-F4=34-3=31=(111)5,F11-F4=89-3=86=(222)6,F13-F5=233-5=228=(444)7,F10-F2=55-1=54=(66)8,F14-F7=377-13=364=(444)9,and F15-F10=610-55=555=(555)*** a result,it is shown that the largest Fibonacci number which can be written as a sum of a repdigit and a Fibonacci number is F15=610=555+55=555+F10.
In this study, we find all Fibonacci and Lucas numbers which can be written as a difference of two repdigits. It is shown that the largest Fibonacci and Lucas numbers which can be written as a difference of two repdig...
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In this study, we find all Fibonacci and Lucas numbers which can be written as a difference of two repdigits. It is shown that the largest Fibonacci and Lucas numbers which can be written as a difference of two repdigits are F-11 = 89 = 111 - 22 and L-18 = 5778 = 6666 - 888, respectively. In particular, the equation F-k = 10(n) - 10(m) has no positive integer solutions (k, m, n).
Let {T-n} (n >= 0) be the sequence of Tribonacci numbers. In this paper, we study the exponential Diophantine equation T-n - 2(x) 3(y) = c, for n, x, y is an element of Z(>= 0). In particular, we show that there...
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Let {T-n} (n >= 0) be the sequence of Tribonacci numbers. In this paper, we study the exponential Diophantine equation T-n - 2(x) 3(y) = c, for n, x, y is an element of Z(>= 0). In particular, we show that there is no integer c with at least six representations of the form T-n - 2(x)3(y).
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