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OPERATIONS ON ELLIPTIC DIVISIBILITY SEQUENCES
Bizim, Osman,Gezer, Betul Korean Mathematical Society 2018 대한수학회보 Vol.55 No.3
In this paper we consider the element-wise (Hadamard) product (or sum) of elliptic divisibility sequences and study the periodic structure of these sequences. We obtain that the element-wise product (or sum) of elliptic divisibility sequences are periodic modulo a prime p like linear recurrence sequences. Then we study periodicity properties of product sequences. We generalize our results to the case of modulo $p^l$ for some prime p > 3 and positive integer l. Finally we consider the p-adic behavior of product sequences and give a generalization of [9, Theorem 4].
Operations on elliptic divisibility sequences
Osman Bizim,Betul Gezer 대한수학회 2018 대한수학회보 Vol.55 No.3
In this paper we consider the element-wise (Hadamard) product (or sum) of elliptic divisibility sequences and study the periodic structure of \ these sequences. We obtain that the element-wise product (or sum) of elliptic divisibility sequences are periodic modulo a prime $p$ like linear recurrence sequences. Then we study periodicity properties of product sequences. We generalize our results to the case of modulo $p^{l}$ for some prime $p>3$ and positive integer $l$. Finally we consider the $p$-adic behavior of product sequences and give a generalization of \cite[Theorem 4] {JS1}.
Rational points on Frey elliptic curves y^2=x^3-n^2x
I. Inam,O. Bizim,I. Cang?l 장전수학회 2007 Advanced Studies in Contemporary Mathematics Vol.14 No.1
In this work, we study the rational points on eliptic curves of the form y2 = x3 n2x over finite fields in a different way, i.e. using only elementary number theory. We calculate the number of rational point sover Fp modulo 8. We show that here are two possible cases wherep 1or 3 (mod 4). In the former case we find a classication of the number of points, while in the later case, we know that here are p + 1 points on the curve by the supersingular curve theory.
The number of representations of positive integers by positive quadratic forms
A. Tekcan,O. Bizim,I. Cangül 장전수학회 2006 Proceedings of the Jangjeon mathematical society Vol.9 No.2
In this paper, we consider the number of representations of pos- itive integers by direct sum of binary quadratic forms F1(x1; y1) = x2 1 + 3x1x2 + 8x2 2 and G1(x1; y1) = 2x2 1 + 3x1x2 + 4x2 2 of discriminant ¡23. We derived some results concerning the modular forms }(¿; Q; 'rs) and their or- ders ord (}(¿; Q; 'rs); i1; ¡0(23)), where Q (Q = F2;G2; F1 ©G1; direct sum of F1 and G1) is the positive de¯nite quadratic form and 'rs is the spherical function of second order with respect to Q. We constructed a basis for the cusp form space S4(¡0(23); 1), and then we obtained some formulas for the number of representations of positive integer n by positive de¯nite quadratic forms F4;G4; F3 © G1; F2 © G2 and F1 © G3 using the elements of the space S4(¡0(23); 1).
The hyperbolic conics in the hyperbolic geometry
Osman Avcioglu,Osman Bizim 장전수학회 2008 Advanced Studies in Contemporary Mathematics Vol.16 No.1
In this work we have aimed to specify basic properties of hyper-bolic conics of uper half plane U and examined each conic at two steps: Firstly we have studied the case where center and focus(es) of the conic are on imaginary axis. Secondly we have transfered our findings to any conic by means of Mob(U). Although there are works on hyperbolic circles in sources such as[2], we also examine hyperbolic circles for readines. We have also seen that each conic family is invariant under the action of Mob(U).
Some relations involving the sums of Fibonacci numbers
A. Tekcan,A. Özkoc,B. Gezer,O. Bizim 장전수학회 2008 Proceedings of the Jangjeon mathematical society Vol.11 No.1
In the first section, we give some preliminaries from Fibonacci numbers which are the sequence given by F0 = 0, F₁= 1 and Fn=Fn-₁+ Fn-₂ for all n ≥ 2. In the second section, we derive explicit formulas for the Fibonacci numbers Fn, F₂n and F₂n+₁. In the fourth section, we give another method for the sums of Fibonacci numbers using the sums of powers of α and β which are the roots of the characteristic equation of Fn=Fn-₁+ Fn-₂. In the last section, we consider the cross-ratio for four consecutive Fibonacci numbers Fn, Fn+₁,Fn+₂ and Fn+₃. We proved that the cross-ratio of Fn, Fn+₁, Fn+₂ and Fn+₃ converges to [수식] as the limit.
Some relations on Lucas numbers and their sums
A. Tekcan,B. Gezer,O. Bizim 장전수학회 2007 Advanced Studies in Contemporary Mathematics Vol.15 No.2
In this paper we consider some algebraic properties of Lucas numbers. In the rst section we give some preliminaries from Lucas and Fibonaccinumbers. In the second section, we derive explicit formulas for the Lucas num-bers Ln ;L2n and L2n +1 . In the third section we consider the sums of Lucasnumbers Ln ;L2n and L2n +1 and also their squares. In the third section, wegive another way for the sums of Lucas numbers. In the last section we consider the cross-ratio of four consecutive Lucas numbers Ln ;Ln +1 ;Ln +2 andLn +3 .
Inam, Ilker,Soydan, Gokhan,Demirci, Musa,BiZim, Osman,Cangul, Ismail Naci Korean Mathematical Society 2007 대한수학회논문집 Vol.22 No.2
In this work, authors considered a result concerning elliptic curves $y^2=x^3+cx$ over $\mathbb{F}_p$ mod 8, given at [1]. They noticed that there should be a slight change at this result. They give counterexamples and the correct version of the result.