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...
详细信息
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.
This article deals with redundant digit expansions with an imaginary quadratic algebraic integer with trace +/- 1 as base and a minimal norm representatives digit set. For w >= 2 it is shown that the width-w non-ad...
详细信息
This article deals with redundant digit expansions with an imaginary quadratic algebraic integer with trace +/- 1 as base and a minimal norm representatives digit set. For w >= 2 it is shown that the width-w non-adjacent form is not an optimal expansion, meaning that it does not minimize the (Hamming) weight among all possible expansions with the same digit set. One main part of the proof uses tools from Diophantine analysis, namely the theory of linear forms in logarithms and the Baker-Davenport reduction method.
Let (U (n) ) (n0) be a nondegenerate binary recurrence sequence with positive discriminant. Let p (1) , . . . , p (s) be fixed prime numbers, b (1) , . . . , b (s) be fixed nonnegative integers, and a (1) , . . . , a ...
详细信息
Let (U (n) ) (n0) be a nondegenerate binary recurrence sequence with positive discriminant. Let p (1) , . . . , p (s) be fixed prime numbers, b (1) , . . . , b (s) be fixed nonnegative integers, and a (1) , . . . , a (t) be positive integers. In this paper, under certain assumptions, we obtain a finiteness result for the solution of the Diophantine equation Moreover, we explicitly solve the equation F (n1) + F (n2) = 2 (z1) + 3 (z2) in nonnegative integers n (1), n (2), z (1), z (2) with z (2) z (1). The main tools used in this work are the lower bound for linear forms in logarithms and the Baker-Davenport reduction method. This work generalizes the recent papers [E. Mazumdar and S.S. Rout, Prime powers in sums of terms of binary recurrence sequences, arXiv:1610.02774] and [C. Bertk, L. Hajdu, I. Pink, and Z. Rabai, linear combinations of prime powers in binary recurrence sequences, Int. J. Number Theory, 13(2):261-271, 2017].
We consider the Catalan equation x(P)-y(q) = 1 in unknowns x, y, p, q, where x, y are taken from an integral domain A of characteristic 0 that is finitely generated as a Z-algebra and p, q > 1 are integers. We give...
详细信息
We consider the Catalan equation x(P)-y(q) = 1 in unknowns x, y, p, q, where x, y are taken from an integral domain A of characteristic 0 that is finitely generated as a Z-algebra and p, q > 1 are integers. We give explicit upper bounds for p and q in terms of the defining parameters of A. Our main theorem is a more precise version of a result of Brindza (1993). Brindza (1987) also gave inexplicit bounds for p and q in the special case that A is the ring of S-integers for some number field K. As part of the proof of our main theorem, we will give a less technical proof for this special case with explicit upper bounds for p and q. (C) 2016 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
In 1939, Erdos and Mahler ['Some arithmetical properties of the convergents of a continued fraction', J. Lond. Math. Soc. (2) 14 (1939), 12-18] studied some arithmetical properties of the convergents of a cont...
详细信息
In 1939, Erdos and Mahler ['Some arithmetical properties of the convergents of a continued fraction', J. Lond. Math. Soc. (2) 14 (1939), 12-18] studied some arithmetical properties of the convergents of a continued fraction. In particular, they raised a conjecture related to continued fractions and Liouville numbers. In this paper, we shall apply the theory of linear forms in logarithms to obtain a result in the direction of this problem.
In this paper, we consider the D(+/- k)-triple {k -/+ 1, k, 4k -/+ 1} and we prove that, if k is not a perfect square then: (1) There is no d such that {k-1, k, 4k-1, d} is a D(k)-quadruple;(2) If {k, k + 1, 4k + 1, d...
详细信息
In this paper, we consider the D(+/- k)-triple {k -/+ 1, k, 4k -/+ 1} and we prove that, if k is not a perfect square then: (1) There is no d such that {k-1, k, 4k-1, d} is a D(k)-quadruple;(2) If {k, k + 1, 4k + 1, d} is a D(-k)-quadruple, then d = 1. This extends a work done by Fujita [13].
In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (n, m, (a), where F-k, is the kth term of the Fibonacci sequence. The proof of our main theorem uses lower bound...
详细信息
In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (n, m, (a), where F-k, is the kth term of the Fibonacci sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms (Baker's theory) and a version of the Baker-Davenport reduction method in diophantine approximation.
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...
详细信息
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.
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 ...
详细信息
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).
暂无评论