#### Notes on Number Theory and Discrete Mathematics

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ISSN / EISSN : 1310-5132 / 2367-8275
Total articles ≅ 335
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Published: 14 June 2022
Notes on Number Theory and Discrete Mathematics, Volume 28; https://doi.org/10.7546/nntdm.2022.28.2.350-375

Abstract:
This paper considers properties of a theorem of Ramanujan to develop properties and algorithms related to cubic equations. The Ramanujan cubics are related to the Cardano cubics and Padovan recurrence relations. These generate cubic identities related to heptagonal triangles and third order recurrence relations, as well as an algorithm for finding the real root of the relevant Ramanujan cubic equation. The algorithm is applied to, and analyzed for, some of the earlier examples in the paper.
, Babeș-Bolyai University
Published: 14 June 2022
Notes on Number Theory and Discrete Mathematics, Volume 28; https://doi.org/10.7546/nntdm.2022.28.2.380-381

, Babeș-Bolyai University,
Published: 14 June 2022
Notes on Number Theory and Discrete Mathematics, Volume 28; https://doi.org/10.7546/nntdm.2022.28.2.376-379

Abstract:
By using the results and methods of [1], we will study the equation \varphi(n) + d(n) = \frac{n}{2} and the related inequalities. The equation \varphi(n) + d^2(n)=2n will be solved, too.
Published: 10 June 2022
Notes on Number Theory and Discrete Mathematics, Volume 28; https://doi.org/10.7546/nntdm.2022.28.2.339-349

Abstract:
In this paper, we define the Hadamard-type k-step Pell sequence by using the Hadamard-type product of characteristic polynomials of the Pell sequence and the k-step Pell sequence. Also, we derive the generating matrices for these sequences, and then we obtain relationships between the Hadamard-type k-step Pell sequences and these generating matrices. Furthermore, we produce the Binet formula for the Hadamard-type k-step Pell numbers for the case that k is odd integers and k ≥ 3. Finally, we derive some properties of the Hadamard-type k-step Pell sequences such as the combinatorial representation, the generating function, and the exponential representation by using its generating matrix.
, Erzincan Binali Yıldırım University,
Published: 10 June 2022
Notes on Number Theory and Discrete Mathematics, Volume 28; https://doi.org/10.7546/nntdm.2022.28.2.318-330

Abstract:
In this work, we investigate the hyperbolic k-Jacobsthal and k-Jacobsthal–Lucas octonions. We give Binet’s Formula, Cassini’s identity, Catalan’s identity, d’Ocagne identity, generating functions of the hyperbolic k-Jacobsthal and k-Jacobsthal–Lucas octonions. Also, we present many properties of these octonions.
Krassimir T. Atanassov
Published: 10 June 2022
Notes on Number Theory and Discrete Mathematics, Volume 28; https://doi.org/10.7546/nntdm.2022.28.2.331-338

Abstract:
A new scheme of 2-Fibonacci sequences is introduced and the explicit formulas for its n-th members are given. For difference of all previous sequences from Fibonacci type, the present 2-Fibonacci sequences are obtained by a new way. It is proved that the new sequences have bases with 48 elements about function 𝜑 and modulo 9.
, , Professional Master Degree Program In Mathematics In National Network – Profmat
Published: 10 June 2022
Notes on Number Theory and Discrete Mathematics, Volume 28; https://doi.org/10.7546/nntdm.2022.28.2.302-317

Abstract:
The Navarrete–Orellana Conjecture states that “given a large prime number a sequence is generated, in such a way that all odd prime numbers, except the given prime, are fixed points of that sequence”. In this work, we formulated a theorem that partially confirms the veracity of this conjecture, more specifically, all prime numbers of a given line segment are fixed points of this sequence.
Published: 12 May 2022
Notes on Number Theory and Discrete Mathematics, Volume 28; https://doi.org/10.7546/nntdm.2022.28.2.286-301

Abstract:
In this study, we deal with the existence of perfect powers which are sum and difference of two balancing numbers. Moreover, as a generalization we explore the perfect squares which are sum and difference of two balancing-like numbers, where balancing-like sequence is defined recursively as $G_{n+1}=AG_n-G_{n-1}$ with initial terms $G_0=0,G_1=1$ for $A \geq 3$.
Published: 12 May 2022
Notes on Number Theory and Discrete Mathematics, Volume 28; https://doi.org/10.7546/nntdm.2022.28.2.281-285

Abstract:
We study equations of type $\sigma(n) = \dfrac{k+1}{k} \cdot n+a,$ where $a\in \{0, 1, 2, 3\},$ where $k$ and $n$ are positive integers, while $\sigma(n)$ denotes the sum of divisors of $n.$
, Ain Shams University
Published: 10 May 2022
Notes on Number Theory and Discrete Mathematics, Volume 28; https://doi.org/10.7546/nntdm.2022.28.2.276-280

Abstract:
Given a positive integer $x$, an addition chain for $x$ is an increasing sequence of positive integers $1=c_0,c_1, \ldots , c_n=x$ such that for each $1\leq k\leq n,$ $c_k=c_i+c_j$ for some $0\leq i\leq j\leq k-1$. In 1937, Scholz conjectured that for each positive integer $x$, $\ell(2^x-1) \leq \ell(x)+ x-1,$ where $\ell(x)$ denotes the minimal length of an addition chain for $x.$ In 1993, Aiello and Subbarao stated the apparently stronger conjecture that there is an addition chain for $2^x-1$ with length equals to $\ell(x)+x-1 .$ We note that the Aiello–Subbarao conjecture is not stronger than the Scholz (also called the Scholz–Brauer) conjecture.