Jacobsthal and Jacobsthal-Lucas numbers with Lehmar property
In this note, we prove that there is no number with the Lehmer property in the sequences of Jaconsthsl or Jacobsthal-Lucas numbers.
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arXiv subjects
Ala'a Al-Kateeb
Publications and source records attributed to Ala'a Al-Kateeb.
Generalized Balancing and Balancing-Lucas numbers
In this paper, we introduce a generalization of Balancing and Balancing-Lucas numbers. We describe some of their properties also we give the related matrix representation and divisibility properties.
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On the number of terms of some families of the ternary cyclotomic polynomials $Φ_{3p_2p_3}$
We study the number of non-zero terms in two specific families of ternary cyclotomic polynomial, we find formulas for the number of terms by writing the cyclotomic polynomial as a sum of smaller sub-polynomials and study the properties of these polynomial.
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Maximum gap in cyclotomic polynomials
Cyclotomic polynomials play fundamental roles in number theory, combinatorics, algebra and their applications. Hence their properties have been extensively investigated. In this paper, we study the maximum gap $g$ (maximum of the differences between any two consecutive exponents). In 2012, it was shown that $g\left( Φ_{p_{1}p_{2}}\right) =p_{1} -1$ for primes $p_{2}>p_{1}$. In 2017, based on numerous calculations, the following generalization was conjectured: $g\left( Φ_{mp}\right) =φ(m)$ for square free odd $m$ and prime $p>m$. The main contribution of this paper is a proof of this conjecture.
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Explicit expression for a family of ternary cyclotomic polynomials
In this paper, we give an explicit expression for a certain family of ternary cyclotomic polynomials: specifically $Φ_{p_{1}p_{2}p_{3}}$, where $p_{1}<p_{2}<p_{3}$ are odd primes such that $p_{2} \equiv1 \mod p_{1}$ and $p_{3} \equiv1 \mod {p_{1}p_{2}}$. As an application of the explicit expressions, we give an exact formula for the number of nonzero terms in the polynomials in the family, which in turn immediately shows that the density (number of non-zeros terms / degree) is roughly inversely proportional to $p_{2}$, when $p_{1}$ is sufficiently large.
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Block Structure of Cyclotomic Polynomials
In this paper, we list several interesting structures of cyclotomic polynomials: specifically relations among blocks obtained by suitable partition of cyclotomic polynomials. We present explicit and self-contained proof for all of them, using a uniform terminology and technique.
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