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Diophantine quadruples for squares of fibonacci and lucas numbers
TLDR
In this article, it was proved that the set n 2Fni1; 2Fn+1, 2F 3 n Fn+1Fni+2, 2 Fn+Ln+2Fni 1; 2 Fni 2 n+1 i F 2 n ) oAbstract:
Let n be an integer. A set of positive integers is said to have the property D(n) if the product of its any two distinct elements increased by n is a perfect square. In this paper, the sets of four numbers represented in terms of Fibonacci numbers with the property D(F 2 n ) and D(L 2 ), where (Fn) is the Fibonacci sequence and (Ln) is the Lucas sequence, are constructed. Among other things, it is proved that the set n 2Fni1; 2Fn+1; 2F 3 n Fn+1Fn+2; 2Fn+1Fn+2Fn+3(2F 2 n+1 i F 2 n ) oread more
Citations
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Journal ArticleDOI
Fibonacci & Lucas numbers, and the golden section: theory and applications, by S. Vajda. Pp 190. £25. 1989. ISBN 0-7458-0715-1 (Ellis Horwood)
Journal ArticleDOI
On Diophantine quintuples
TL;DR: The Diophantine m-tuple with the property that the product of any two distinct elements increased by 1 is a square of a rational number was shown to have the same property as the Diophantus quadruple in this paper.
The problem of diophantus and davenport for gaussian integers
Andrej Dujella,A. Dujella +1 more
TL;DR: In this article, it is proved that if a Gaussian integer z is not representable as a difference of the squares of two Gaussian integers, then there does not exist a quadruple with the property D(z).
Some polynomial formulas for Diophantine quadruples
TL;DR: The problem of finding four positive rational numbers such that the product of any two of them increased by 1 is a perfect square was studied by Diophantus of Alexandria as discussed by the authors.
Book ChapterDOI
On the Exceptional Set in the Problem of Diophantus and Davenport
TL;DR: In this article, it was shown that if d is a positive integer such that d has the property of Diophantus of Alexandria, then d has to be 120, and if d has a property that any two positive integers increased by 1 is a square of a rational number then d must be 120.
References
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Journal ArticleDOI
Generating functions for powers of a certain generalised sequence of numbers
Journal ArticleDOI
Fibonacci & Lucas numbers, and the golden section: theory and applications, by S. Vajda. Pp 190. £25. 1989. ISBN 0-7458-0715-1 (Ellis Horwood)
Journal ArticleDOI
Sets in which $xy+k$ is always a square
TL;DR: In this article, it was shown that if k = 2 (mod 4), then there does not exist a Pk-set of size 4, and that the P_,-set {1, 2. 5} cannot be extended.
Journal ArticleDOI
Generalization of a problem of Diophantus
TL;DR: In this article, it was shown that for any integer n, there is an infinite number of sets of four natural numbers with Diophantus' property D(e2) of order n, and that such a set can be extended to a set {a, b, c, d} with the same property, if b is not a perfect square.