Browse > Article
http://dx.doi.org/10.11568/kjm.2020.28.3.539

THE EXTENDIBILITY OF DIOPHANTINE PAIRS WITH PROPERTY D(-1)  

Park, Jinseo (Department of Mathematics Education Catholic Kwandong University)
Publication Information
Korean Journal of Mathematics / v.28, no.3, 2020 , pp. 539-554 More about this Journal
Abstract
A set {a1, a2, …, am} of m distinct positive integers is called a D(-1)-m-tuple if the product of any distinct two elements in the set decreased by one is a perfect square. In this paper, we find a solution of Pellian equations which is constructed by D(-1)-triples and using this result, we prove the extendibility of D(-1)-pair with some conditions.
Keywords
Diophantine m-tuple; Fibonacci number; Pellian equation;
Citations & Related Records
연도 인용수 순위
  • Reference
1 K. S. Kedlaya, Solving constrained Pell equations, Math. Comp. 67 (1998), 833-842.   DOI
2 A. Baker, G. Wustholz, Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19-62.
3 E. Brown, Sets in which xy + k is always a square, Math. Comp. 45 (1985), 613-620.   DOI
4 L. E. Dickson, History of the Theory of Numbers Vol.2, Chelsea, New York, 1966.
5 A. Dujella, Complete solution of a family of simultaneous Pellian equations, Acta Math. Inform. Univ. Ostraviensis 6 (1998), no.1, 59-67.
6 A. Dujella, A parametric family of elliptic curves, Acta Arith. 94 (2000), no. 1, 87-101.   DOI
7 A. Dujella,Effective solution of the D(-1)-quadurple conjecture, Acta Arith. 128 (2007), no. 4, 319-338.   DOI
8 A. Dujella, C. Fuchs, Complete solution of a problem of Diophantus and Euler, J. London Math. Soc. (2) 71 (2005), 33-52.   DOI
9 A. Filipin, Y. Fujita, The relative upper bound for the third element in a D(-1)-quadruple, Math. Commun. 17 (2012), no. 1, 13-19.
10 Y. Fujita, The extensibility of Diophantine pair {k-1, k+1}, J. Number Theory 128 (2008), no. 2, 323-353.   DOI
11 Y. Fujita, The Hoggatt-Bergum conjecture on D(-1)-triples {$F_{2k+1}$, $F_{2k+3}$, $F_{2k+5}$} and integer points on the attached elliptic curves, Rocky Mountain J. Math. 39 (2009), no. 6, 1907-1932.   DOI
12 B. He, A. Togbe, V. Ziegler, On the D(-1)-triple {1, $k^2$ + 1, $k^2$ + 2k + 2} and its unique D(1)-extension, J. Number Theory 131 (2011), 120-137.   DOI
13 B. He, A. Togbe, V. Ziegler, There is no Diophantine quintuple, Trans. Amer. Math. Soc., 371 (2019), 6665-6709.   DOI