Some remarks regarding l-elements defined in algebras obtained by the Cayley–Dickson process

Flaut, Cristina; Savin, Diana

doi:10.1016/j.chaos.2018.11.002

Cited by 4 publications

(5 citation statements)

References 17 publications

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“…Proof In the paper, 21 we obtained that the norm of the

n

l -

quaternion is

n ((), A_{n}) = ((), l^{2} + 2) a_{2 n + 3}

. Applying Proposition 4.4. ii), it results that

n ((), A_{n}) \equiv

2 mod l. Since l is odd, we have that

n ((), A_{n}) \neq \overset{0}{true}

{double-struckZ}_{l}

, for all

n \in double-struckN

.…”

Section: Applications Of Some Special Number Sequences and Quaternion Elementsmentioning

confidence: 93%

“…Proposition Let

{((), a_{n})}_{n \geq 0}

be the sequence previously defined 21 . The following relations hold: i) If

d false| n

, then

a_{d} false| a_{n}

. ii)

a_{m + n} = a_{m} a_{n + 1} + a_{m - 1} a_{n} . □

□…”

Section: Applications Of Some Special Number Sequences and Quaternion Elementsmentioning

confidence: 99%

“…Proposition Let

{false(a_{n} false)}_{n \geq 0}

be the sequence previously defined 21 . Then, the following relations are true: i)

a_{n} + a_{n + 4} = (l^{2} + 2) a_{n + 2};

ii)

a_{n} + a_{n + 8} = [{(l^{2} + 2)}^{2} - 2] a_{n + 4};

iii)

a_{n} + a_{n + 2^{k}} = M_{k} a_{n + 2^{k - 1}}, k \geq 3,

where

M_{k} = \overset{\underset{⏟}{([], {((), {((), {((), l^{2} + 2)}^{2} - 2)}^{2} . . . - 2)}^{2} - 2)}}{true}

.…”

Section: Applications Of Some Special Number Sequences and Quaternion Elementsmentioning

confidence: 99%

“…Proof. In the paper, 21 we obtained that the norm of the nth l− quaternion is n (A n ) = ( l 2 + 2 ) a 2n+3 . Applying Proposition 4.4. ii), it results that n (A n ) ≡ 2 mod l. Since l is odd, we have that n (A n ) ≠0 in Z l , for all n ∈ N. Thus, all the n th l− quaternions are invertible in the quaternion algebra H Z l (−1, −1).…”

mentioning

confidence: 99%

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Properties and applications of some special integer number sequences

Flaut

Savin

Zaharia

2020

Math Methods in App Sciences

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“…Proof In the paper, 21 we obtained that the norm of the

n

l -

quaternion is

n ((), A_{n}) = ((), l^{2} + 2) a_{2 n + 3}

. Applying Proposition 4.4. ii), it results that

n ((), A_{n}) \equiv

2 mod l. Since l is odd, we have that

n ((), A_{n}) \neq \overset{0}{true}

{double-struckZ}_{l}

, for all

n \in double-struckN

.…”

Section: Applications Of Some Special Number Sequences and Quaternion Elementsmentioning

confidence: 93%

“…Proposition Let

{((), a_{n})}_{n \geq 0}

be the sequence previously defined 21 . The following relations hold: i) If

d false| n

, then

a_{d} false| a_{n}

. ii)

a_{m + n} = a_{m} a_{n + 1} + a_{m - 1} a_{n} . □

□…”

Section: Applications Of Some Special Number Sequences and Quaternion Elementsmentioning

confidence: 99%

“…Proposition Let

{false(a_{n} false)}_{n \geq 0}

be the sequence previously defined 21 . Then, the following relations are true: i)

a_{n} + a_{n + 4} = (l^{2} + 2) a_{n + 2};

ii)

a_{n} + a_{n + 8} = [{(l^{2} + 2)}^{2} - 2] a_{n + 4};

iii)

a_{n} + a_{n + 2^{k}} = M_{k} a_{n + 2^{k - 1}}, k \geq 3,

where

M_{k} = \overset{\underset{⏟}{([], {((), {((), {((), l^{2} + 2)}^{2} - 2)}^{2} . . . - 2)}^{2} - 2)}}{true}

.…”

Section: Applications Of Some Special Number Sequences and Quaternion Elementsmentioning

confidence: 99%

mentioning

confidence: 99%

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Properties and applications of some special integer number sequences

Flaut

Savin

Zaharia

2020

Math Methods in App Sciences

Self Cite

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show abstract

“…Detailed information on properties of the elements of matrix Q can be found in [5,6]. If necessary actions are taken for the second side of equation (31) and by using the equalities…”

Section: The Fibonacci Matrixmentioning

confidence: 99%