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Why Quadratic Diophantine Equations? Read chapter Read first chapter

2015 | OriginalPaper | Chapter

6. Diophantine Representations of Some Sequences

Authors : Titu Andreescu, Dorin Andrica

Published in: Quadratic Diophantine Equations

Publisher: Springer New York

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Abstract

In 1900, David Hilbert asked for an algorithm to decide whether a given Diophantine equation is solvable or not and put this problem tenth in his famous list of 23.

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previous chapter Equations Reducible to Pell’s Type Equations
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Literature
20.
Andreescu, T., Andrica, D.: Equations with solution in terms of Fibonacci and Lucas sequences. An. Şt. Univ. Ovidius Constanţa Ser. Mat. 22(3), 5–12 (2014)MATHMathSciNet
21.
Andreescu, T., Andrica, D.: Number Theory. Structures, Examples and Problems. Birkhäuser, Boston (2009)MATH
25.
27.
Andrica, D., Buzeteanu, Ş.: On the reduction of a linear recurrence of r th order. Fibonacci Q. 23(1), 81–84 (1985)MATHMathSciNet
28.
Andrica, D., Buzeţeanu, Ş.: Reducerea unei recurenţe liniare de ordinul doi şi consecinţe ale acesteia (Romanian). Gazeta Matematică seria A (2) 3(3–4), 148–152 (1982)
35.
Bastida, J.R.: Quadratic properties of a linearly recurrent sequence. In: Proceedings of the 10th Southeastern Conference on Combinatorics, Graph Theory and Computing. Winnipeg (Utilitas Mathematica) (1979)
36.
Bastida, J.R., DeLeon. M.R.: A quadratic property of certain linearly recurrent sequences. Fibonacci Q. 19(2), 144–146 (1981)
47.
Buzeţeanu, Ş.: Grade de efectivitate în teoria calculului. Aspecte recursive, analitice şi combinatoriale (Romanian). Teză de doctorat, Universitatea din Bucureşti (1988)
60.
61.
83.
Halici, S.: On some inequalities and Hankel matrices involving Pell, Pell-Lucas numbers. Math. Rep. 15(65/1), 1–10 (2013)MathSciNet
90.
Hoggatt, Jr., V.E.: Fibonacci and Lucas Numbers. The Fibonacci Association, Santa Clara (1969)MATH
91.
Hoggatt, V.E., Bicknell-Johnson, M.: A primer for the Fibonacci XVII: generalized Fibonacci numbers satisfying \(u_{n+1}u_{n-1} - u_{n}^{2} = \pm 1\). Fibonacci Q. 16(2), 130–137 (1978)MATHMathSciNet
92.
Horadam, A.F.: Geometry of a generalized Simson’s formula. Fibonacci Q. 20(2), 164–168 (1982)MATH
96.
Jones, J.P.: Diophantine representation of Fibonacci numbers. Fibonacci Q. 13(1), 84–88 (1975)MATH
97.
Jones, J.P.: Diophantine representation of the Lucas numbers. Fibonacci Q. 14(2), 134 (1976)MATH
98.
Jones, J.P.: Representation of solutions of Pell equations using Lucas sequences. Acta Academiae Pedagogicae Agriensis. Sectio Matematicae 30, 75–86 (2003)MATH
102.
Keskin, R., Demiturk, N.: Solutions to some Diophantine equations using generalized Fibonacci and Lucas sequences. Ars Combinatoria CXI, 161–179 (2013)
104.
Knuth, D.E.: The Art of Computer Programming. Fundamental Algorithms, vol. I. Addison-Wesley, Boston (1975)
105.
106.
Koshy, T.: Pell and Pell-Lucas Numbers with Applications. Springer, NewYork (2015)
114.
Li, Z.: The positive integer solutions of ax 2 − by 2 = c. Octogon Math. Mag. 10(1), 348–349 (2002)
133.
Matiyasevich, Yu.: Enumerable sets are Diophantine. Soviet Math. Doklady 11, 354–358 (1970)MATH
134.
135.
McLaughlin, R.: Sequences. some properties by Matrix methods. Math. Gaz. 64(430), 281–282 (1980)
150.
Mordell, L.J.: Diophantine equations. Academic, London (1969)MATH
178.
183.
187.
Roy, S.: What’s the next Fibonacci numbers. Math. Gaz. 63(429), 189–190 (1980)CrossRef
188.
Savin, D.: About a Diophantine equation. An. Şt. Univ. Ovidius Constanţa Ser. Mat. 17(3), 241–250 (2009)MATHMathSciNet
200.
Sierpinski, W.: 250 Problems in Elementary Number Theory. American Elsevier, New York/ PWN, Warszawa (1970)MATH
202.
Silvester, J.R.: Fibonacci properties by matrix methods. Math. Gaz. 63(425), 188–191 (1979)MATHCrossRef
217.
230.
Zeckendorf, E.: Représentation des nombres naturels par une somme de nombres de Fibonacci ou des nombres de Lucas. Bull. Soc. R. Sci. Liège 41, 179–182 (1972)MATHMathSciNet
Metadata
Title
Diophantine Representations of Some Sequences
Authors
Titu Andreescu
Dorin Andrica
Copyright Year
2015
Publisher
Springer New York
Book
Quadratic Diophantine Equations

Print ISBN: 978-0-387-35156-8

Electronic ISBN: 978-0-387-54109-9

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-0-387-54109-9_6

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