We give a separation bound for the complex roots of a trinomial f is an element of Z[X]. The logarithm of the inverse of our separation bound is polynomial in the size of the sparse encoding of f;in particular, it is ...
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We give a separation bound for the complex roots of a trinomial f is an element of Z[X]. The logarithm of the inverse of our separation bound is polynomial in the size of the sparse encoding of f;in particular, it is polynomial in log(deg f). It is known that no such bound is possible for 4-nomials (polynomials with 4 monomials). For trinomials, the classical results (which are based on the degree of f rather than the number of monomials) give separation bounds that are exponentially worse. As an algorithmic application, we show that the number of real roots of a trinomial f can be computed in time polynomial in the size of the sparse encoding of f. The same problem is open for 4-nomials. (C) 2019 Elsevier Ltd. All rights reserved.
Quadruples (a, b, c, d) of positive integers a < b < c < d with the property that the product of any two of them is one more than a perfect square are studied. Improved lower and upper bounds for the entries ...
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Quadruples (a, b, c, d) of positive integers a < b < c < d with the property that the product of any two of them is one more than a perfect square are studied. Improved lower and upper bounds for the entries b and c are established. As an application of these results, a bound for the number of such quadruples is obtained.
We will give upper bounds for the number of integral solutions to quartic Thue equations. Our main tool here is a logarithmic curve phi(x, y) that allows us to use the theory of linear forms in logarithms. This paper ...
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We will give upper bounds for the number of integral solutions to quartic Thue equations. Our main tool here is a logarithmic curve phi(x, y) that allows us to use the theory of linear forms in logarithms. This paper improves the results of the author's earlier work with Okazaki [The quartic Thue equations, J. Number Theory 130(1) (2010) 40-60] by giving special treatments to forms with respect to their signature.
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.
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.
In this paper we consider an analogue of the problem of Erdos and Woods for arithmetic progressions. A positive answer follows from the abe conjecture. Partial results are obtained unconditionally.
In this paper we consider an analogue of the problem of Erdos and Woods for arithmetic progressions. A positive answer follows from the abe conjecture. Partial results are obtained unconditionally.
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.
Assuming a weaker form of the Riemann hypothesis for Dedekind zeta functions by allowing Siegel zeros, we extend a classical result of Cramer on the number of primes in short intervals to prime ideals of the ring of i...
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Assuming a weaker form of the Riemann hypothesis for Dedekind zeta functions by allowing Siegel zeros, we extend a classical result of Cramer on the number of primes in short intervals to prime ideals of the ring of integers in cyclotomic extensions with norms belonging to such intervals. The extension is uniform with respect to the degree of the cyclotomic extension. Our approach is based on the arithmetic of cyclotomic fields and analytic properties of their Dedekind zeta functions together with a lower bound for the number of primes over progressions in short intervals subject to similar assumptions. Uniformity with respect to the modulus of the progression is obtained and the lower bound turns out to be best possible, apart from constants, as shown by the Brun-Titchmarsh theorem. (C) 2016 Elsevier Inc. All rights reserved.
We prove that, for positive integers a, b, c and d with c not equal d, a > 1, b > 1, the number of simultaneous solutions in positive integers to ax(2) - cz(2) = 1, by(2) - dz(2) = 1 is at most two. This result ...
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We prove that, for positive integers a, b, c and d with c not equal d, a > 1, b > 1, the number of simultaneous solutions in positive integers to ax(2) - cz(2) = 1, by(2) - dz(2) = 1 is at most two. This result is the best possible one. We prove a similar result for the system of equations x(2) - ay(2) = 1, z(2) - bx(2) = 1. (c) 2006 Elsevier Inc. All rights reserved.
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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