About Special Elements in Quaternion Algebras Over Finite Fields

Savin, Diana

doi:10.1007/s00006-016-0718-2

Cited by 16 publications

(18 citation statements)

References 10 publications

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“…The above Theorem generalized Theorems 2.2, 2.4, 2.6 from [Ko,Oz,Si;17] to D−polynomials. Using ideas from the above mentioned paper, where were presented cyclic codes obtained from Fibonacci polynomials, in the following, for particular cases, we will present some properties of cyclic codes obtained from D−polynomials.…”

Section: An Application In the Coding Theorymentioning

confidence: 71%

“…In [Ba,Pr;09], [St;06] and [Ko,Oz,Si;17] were presented some applications of the Fibonacci elements in the Coding Theory. In the following, we will give applications of other special numbers in this domain.…”

Section: An Application In the Coding Theorymentioning

confidence: 99%

“…For π = 3, we have l f ib (π) = 8, β f ib (π) = 2, for π = 5, we get l f ib (π) = 20, β f ib (π) = 4, for π = 7, we have l f ib (π) = 16, β f ib (π) = 2, for π = 11, we obtain l f ib (π) = 10, β f ib (π) = 1, for π = 13, we have l f ib (π) = 28, β f ib (π) = 4. (Also you can see [Ko,Oz,Si;17]) ii) For a = 2, we get the Pell numbers. For π = 3, we have l pell (π) = 8, β pell (π) = 2.…”

Section: An Application In the Coding Theorymentioning

confidence: 99%

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Some special number sequences obtained from a difference equation of degree three

Flaut

Savin

2018

Chaos, Solitons & Fractals

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Abstract. In this paper we present applications of special numbers obtained from a difference equation of degree three. One of these applications is in the Coding Theory, since some of these numbers can be used to built cyclic codes with good properties (MDS codes). In another particular case of these difference equation of degree three, we obtain the generalized Pell-Fibonacci-Lucas numbers, which were extended to the generalized quaternion algebras. Using properties of these elements, we can define a set with an interesting algebraic structure, namely an order of a generalized rational quaternion algebra.

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Section: An Application In the Coding Theorymentioning

confidence: 71%

Section: An Application In the Coding Theorymentioning

confidence: 99%

Section: An Application In the Coding Theorymentioning

confidence: 99%

See 1 more Smart Citation

Some special number sequences obtained from a difference equation of degree three

Flaut

Savin

2018

Chaos, Solitons & Fractals

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“…In 1975, Sallé [22] proved that z(n) ≤ 2n for all positive integers n. This is the sharpest upper bound for z(n), since, for example, z(6) = 12 and z(30) = 60. In the case of a prime number p, one has better upper bound z(p) ≤ p + 1 (Savin [23] proved that, for primes p ≡ 13, 17 (mod 20) holds z(p) | (p + 1)/2, so z(p) ≤ (p + 1)/2). Equality z(p) = p + 1 is achieved for some prime numbers.…”

Section: Introductionmentioning

confidence: 99%

On Diophantine Equations Related to Order of Appearance in Fibonacci Sequence

Trojovský

2019

Mathematics

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Let F n be the nth Fibonacci number. Order of appearance z ( n ) of a natural number n is defined as smallest natural number k, such that n divides F k . In 1930, Lehmer proved that all solutions of equation z ( n ) = n ± 1 are prime numbers. In this paper, we solve equation z ( n ) = n + ℓ for | ℓ | ∈ { 1 , … , 9 } . Our method is based on the p-adic valuation of Fibonacci numbers.

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“…In 2016, Savin studied special Fibonacci quaternions and special generalized Fibonacci‐Lucas quaternions

G_{n}^{p, q} = g_{n}^{p, q} + i g_{n + 1}^{p, q} + j g_{n + 2}^{p, q} + k g_{n + 3}^{p, q}, i^{2} = j^{2} = k^{2} = i j k = - 1,

where

g_{n}^{p, q} = p F_{n} + q L_{n + 1}

in quaternion algebras over finite fields.…”

Section: Introductionmentioning

confidence: 99%

On metallic ratio in

Akbiyik

Yüce

2019

Math Methods in App Sciences

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Metallic ratio is a root of the simple quadratic equation x2 = kx + 1 for k is any positive integer which is the characteristic equation of the recurrence relation of k‐Fibonacci (k‐Lucas) numbers. This paper is about the metallic ratio in Zp. We define k‐Fibonacci and k‐Lucas numbers in Zp, and we show that metallic ratio can be calculated in Zp if and only if p≡ ± 1 mod (k2 + 4), which is the generalization of the Gauss reciprocity theorem for any integer k. Also, we obtain that the golden ratio, the silver ratio, and the bronze ratio, the three together, can be calculated in Z79 for the first time. Moreover, we introduce k‐Fibonacci and k‐Lucas quaternions with some algebraic properties and some identities for them.

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About Special Elements in Quaternion Algebras Over Finite Fields

Abstract: Abstract. In this paper we study special Fibonacci quaternions and special generalized Fibonacci-Lucas quaternions in quaternion algebras over finite fields.

Cited by 16 publications

References 10 publications

Some special number sequences obtained from a difference equation of degree three

Some special number sequences obtained from a difference equation of degree three

On Diophantine Equations Related to Order of Appearance in Fibonacci Sequence

On metallic ratio in

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