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Let $${f(x, k, d) = x(x + d)\cdots(x + (k - 1)d)}$$ f ( x , k , d ) = x ( x + d ) ? ( x + ( k - 1 ) d ) be a polynomial with $${k \geq 2}$$ k ≥ 2 , $${d \geq 1}$$ d ≥ 1 . We consider the Diophantine equation $${\prod_{i = 1}^{r}...
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Let $${f(x, k, d) = x(x + d)\cdots(x + (k - 1)d)}$$ f ( x , k , d ) = x ( x + d ) ? ( x + ( k - 1 ) d ) be a polynomial with $${k \geq 2}$$ k ≥ 2 , $${d \geq 1}$$ d ≥ 1 . We consider the Diophantine equation $${\prod_{i = 1}^{r} f(x_i, k_i, d) = y^2}$$ ∏ i = 1 r f ( x i , k i , d ) = y 2 , which is inspired by a question of Erd?s and Graham [4, p. 67]. Using the theory of Pellian equation, we give infinitely many (nontrivial) positive integer solutions of the above Diophantine equation for some cases.
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In this paper we consider the family of systems (2c + 1)U~2 — 2cV~2 = μ and (c — 2)U~2 — cZ~2 = — 2μ of relative Pellian equations, where the parameter c and the root of unity μ are integers in the same imaginary quadratic ...
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In this paper we consider the family of systems (2c + 1)U~2 — 2cV~2 = μ and (c — 2)U~2 — cZ~2 = — 2μ of relative Pellian equations, where the parameter c and the root of unity μ are integers in the same imaginary quadratic number field K = Q((-d)~(1/2)). We show that for |c| ≥ 3 only certain values of μ yield solutions of this system, and solve the system completely for |c| ≥ 1544686. Furthermore we will consider the related relative Thue equation X~4 - 4cX~3Y + (6c + 2)X~2Y~2 + 4cXY~3 + Y~4 = μ and solve it by the method of Tzanakis under the same assumptions.
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It this paper, we study the existence of Diophantine quadruples with property D(z) in the ring Z[root d], where d is such that the Pellian equation x(2) - dy(2) = +/- 2 is solvable. This existence is characterized by the represent...
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It this paper, we study the existence of Diophantine quadruples with property D(z) in the ring Z[root d], where d is such that the Pellian equation x(2) - dy(2) = +/- 2 is solvable. This existence is characterized by the representability of z as a difference of two squares.
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Abstract Using the theory of Pellian equations, we show that the Diophantine equations z2=f(x)2±f(y)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{a...
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Abstract Using the theory of Pellian equations, we show that the Diophantine equations z2=f(x)2±f(y)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z^2=f(x)^2 \pm f(y)^2$$\end{document} have infinitely many nontrivial integer solutions (x,?y,?z) for three classes of polynomials f(x)∈Z[x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x)\in {\mathbb {Z}}[x]$$\end{document} of any degree n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document}, which extend the results of He et al. (Bull Aust Math Soc 82(2):187–204, 2010).
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In this paper, we study the problem of determining the elements in the rings of integers of quadratic fields Q(root d) which are representable as a difference of two squares. The complete solution of the problem is obtained for in...
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In this paper, we study the problem of determining the elements in the rings of integers of quadratic fields Q(root d) which are representable as a difference of two squares. The complete solution of the problem is obtained for integers d which satisfy conditions given in terms of solvability of certain Pellian equations.
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We prove that for every integer n, there exist infinitely many D(n)-triples which are also D(t)-triples for t is an element of Z with n not equal t. We also prove that there are infinitely many D(-1)-triples in Z[i] which are also...
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We prove that for every integer n, there exist infinitely many D(n)-triples which are also D(t)-triples for t is an element of Z with n not equal t. We also prove that there are infinitely many D(-1)-triples in Z[i] which are also D(n)-triple in Z[i] for two distinct n's other than n = -1 and these triples are not equivalent to any triple with the property D(1).
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In this paper, we consider the problem of existence of Diophantine m-tuples which are (not necessarily consecutive) elements of an arithmetic progression. We show that for n ≥ 3 there does not exist a Diophantine quintuple {a, b, ...
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In this paper, we consider the problem of existence of Diophantine m-tuples which are (not necessarily consecutive) elements of an arithmetic progression. We show that for n ≥ 3 there does not exist a Diophantine quintuple {a, b, c, d, e} such that a ≡ b ≡ c ≡ d ≡ e(mod n). On the other hand, for any positive integer n there exist infinitely many Diophantine triples {a, b, c} such that a ≡ b ≡ c ≡ 0(mod n).
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摘要 :It this paper, we study the existence of Diophantine quadruples with property D(z) in the ring $\mathbb {Z}[\sqrt {d}]$ , where d is such that the Pellian equation x 2−dy 2=±2 is solvable. This existence is characterized by the ...
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It this paper, we study the existence of Diophantine quadruples with property D(z) in the ring $\mathbb {Z}[\sqrt {d}]$ , where d is such that the Pellian equation x 2−dy 2=±2 is solvable. This existence is characterized by the representability of z as a difference of two squares.
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A Ducci sequence is a sequence of integer the aximal period of such sequences for given. We prove a new upper bound in the case where is a power of a prime p equivalent to 5 (mod 8) for which is a primitive root and the Pellian eq...
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A Ducci sequence is a sequence of integer the aximal period of such sequences for given. We prove a new upper bound in the case where is a power of a prime p equivalent to 5 (mod 8) for which is a primitive root and the Pellian equation has no solutions in odd integers x and y.
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