Volume 25 ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4
Sums of reciprocals of integers
Original research paper. Pages 1–7
Simon Davis
Full paper (PDF, 182 Kb) | Abstract
The sums of reciprocals are demonstrated to diverge for infinite sequences consisting of arbitrarily long arithmetic progressions. It is demonstrated that there may exist sequences that do not include arithmetic progressions of arbitrary length that yield divergent sums.
Improving the estimates for a sequence involving prime numbers
Original research paper. Pages 8–13
Christian Axler
Full paper (PDF, 146 Kb) | Abstract
Based on new explicit estimates for the prime counting function, we improve the currently known estimates for the particular sequence Cn = npn − ∑k ≤ n pk, n ≥ 1, involving the prime numbers.
New methods for obtaining new families of congruent numbers
Original research paper. Pages 14–24
Hamid Reza Abdolmalki and Farzali Izadi
Full paper (PDF, 191 Kb) | Abstract
In this note, we introduce elementary methods for obtaining new families of congruent numbers (CNs). By our methods, we can produce other CNs when one or two CNs are given. Also, we use some of the Pell equations (PEs) for getting some families of CNs. Up to now, it is not exactly determined which prime numbers of the form p = 8k + 1 are CNs. Among other things, we also introduce two simple methods to find some CNs of the forms p ≡ 1 (mod 8) and 2p where p is a prime number. By non-CNs and our methods, we also obtain some Diophantine equations (especially of degree 4), which have no positive solutions. In the end, we obtain a result on Heron triangles.
Larger families of solutions to some Diophantine equations
Original research paper. Pages 25–31
Lyes Ait-Amrane
Full paper (PDF, 138 Kb) | Abstract
In this paper, we give three identities involving the Lucas sequences of the first kind and of the second kind in order to obtain infinite families of solutions to some diophantine equations. Some of these families are new and the others are larger than those known until now.
A note on Euler’s totient function
Original research paper. Pages 32–35
József Sándor
Full paper (PDF, 157 Kb) | Abstract
We prove by elementary arguments that the inequalities ϕ(2k + 1) > 2k −1 and ϕ(2m + 1) < 2m − 1 both have infinitely many solutions.
Minimal sets of shifted values of the Euler totient function
Original research paper. Pages 36–47
Martin Kreh and Katrin Neuenstein
Full paper (PDF, 193 Kb) | Abstract
In this article we determine the minimal set for some sets of natural numbers. The concept of minimal sets (in the context of natural numbers) appeared first in an article of Shallit, who determined, among others, the minimal set of the primes. By now, there are several articles about minimal sets. In this article we will expand results of Baoulina, Kreh and Steuding, who determined the minimal set of the sets φ(ℕ) and φ(ℕ) + 3. To this end, we will determine the minimal set of the sets φ(ℕ) + a for 1 ≤ a ≤ 5.
New form of the Newton’s binomial theorem
Original research paper. Pages 48–49
Mladen V. Vassilev-Missana
Full paper (PDF, 119 Kb) | Abstract
A new version of the Newton’s binomial theorem has been proposed and proved in the paper.
Inequalities between the arithmetic functions φ, ψ and σ. Part 1
Original research paper. Pages 50–53
Krassimir T. Atanassov and József Sándor
Full paper (PDF, 149 Kb) | Abstract
For three of the basic arithmetic functions φ, ψ and σ are proved the inequalities ψ(n)n > σ(n)φ(n) and σ(n)n < ψ(n)σ(n) for each natural number n ≥ 2.
Partitions generated by Mock Theta Functions ρ(q), σ(q) and ν(q) and relations with partitions into distinct parts
Original research paper. Pages 54–74
Alessandro Bagatini, Marília Luiza Matte and Adriana Wagner
Full paper (PDF, 252 Kb) | Abstract
From two-line matrix interpretations of Mock Theta Functions ρ(q), σ(q) and ν(q) introduced in [5], we have obtained identities for the partitions generated by their respective general terms, whose proofs are done in a completely combinatorial way. We have also obtained relations between partitions into two colours generated by ρ(q) and σ(q), and also by ν(q).
Complete solving the quadratic equation mod 2n
Original research paper. Pages 75–83
S. M. Dehnavi, M. R. Mirzaee Shamsabad and A. Mahmoodi Rishakani
Full paper (PDF, 213 Kb) | Abstract
Quadratic functions have applications in cryptography. In this paper, we investigate the modular quadratic equation ax2 + bx + c = 0 (mod 2n), and provide a complete analysis of it. More precisely, we determine when this equation has a solution and in the case that it has a solution, we give not only the number of solutions, but also the set of solutions, in O(n) time. One of the interesting results of our research is that, if this equation has a solution, then the number of solutions is a power of two. Most notably, as an application, we characterize the number of fixed-points of quadratic permutation polynomials over ℤ2n, which are used in symmetric cryptography.
On Terai’s exponential equation with two finite integer parameters
Original research paper. Pages 84–107
Takafumi Miyazaki
Full paper (PDF, 247 Kb) | Abstract
Let r be an integer with r > 1, and m be an even positive integer. Define integers A and B by the equation A + B √−1 = (m + √−1)r. It is proven by F. Luca in 2012 that the equation |A|x + |B|y = (m2 + 1)z does not hold for any triple (x, y, z) of positive integers not equal to (2, 2, r), whenever r or m exceeds some effectively computable absolute constant. In our previous work, we estimated this constant explicitly. Here that estimate is substantially improved.
Almost balancing, triangular and square triangular numbers
Original research paper. Pages 108–121
Ahmet Tekcan
Full paper (PDF, 196 Kb) | Abstract
In this work, we derive some new algebraic relations on all almost balancing numbers (of first and second type) and triangular (and also square triangular) numbers.
Averages of the Dirichlet convolution of the binary digital sum
Original research paper. Pages 122–127
Teerapat Srichan
Full paper (PDF, 144 Kb) | Abstract
We derive some averages of the Dirichlet convolution of the binary digital sum s2(n), the sum of digits ofthe expansion of n in base 2. The Trollope–Delange formula is used in our proof. It provides an explicit asymptotic formula for the total number of digits ‘1’ in the binary expansions of the integers between 1 and n − 1 in term of the continuous, nowhere differentiable Takagi function. Moreover, we also extend the result to averages of the k-th convolution of the binary digital sum, for k ≥ 2.
Bijective proofs involving chromatic overpartitions
Original research paper. Pages 128–136
Mateus Alegri
Full paper (PDF, 171 Kb) | Abstract
In this paper, our aim is to provide two bijective proofs for identities involving what we call chromatic overpartitions, which is a generalization of the well-known overpartitions class. For this purpose we will give the mathematical definitions of chromatic overpartitions, providing their respective generating functions.
On dual Horadam octonions
Original research paper. Pages 137–149
Nayil Kılıç
Full paper (PDF, 294 Kb) | Abstract
In this paper, we introduce the dual Horadam octonions, we give the Binet formula, generating function, exponential generating function, summation formula, Catalan’s identity, Cassini’s identity and d’Ocagne’s identity of dual Horadam octonions. Employing these results, we present the Binet formula, generating function, summation formula, Catalan, Cassini and d’Ocagne identities for dual Fibonacci, dual Lucas, dual Jacobsthal, dual Jacobsthal–Lucas, dual Pell and dual Pell–Lucas octonions. So we generalize results that were obtained earlier by scientists. Finally, we introduce the matrix generator for dual Horadam octonions and this generator gives the Cassini formula for the dual Horadam octonions.
Power GCD and power LCM matrices defined on GCD-closed sets over unique factorization domains
Original research paper. Pages 150–166
Y. A. Awad, T. Kadri and R. H. Mghames
Full paper (PDF, 225 Kb) | Abstract
Let T = {t1, t2, …, tm} be a well ordered set of m distinct positive integers with t1 < t2 < … < tm. The GCD matrix on T is defined as (T)m×m = (ti, tj), where (ti, tj) is the greatest common divisor of ti and tj , and the power GCD matrix on T is (Tr)m×m = (ti, tj)r, where r is any real number. The LCM matrix on T is defined as [T]m×m = [ti, tj], where [ti, tj] is the least common multiple of ti and tj, and the power LCM matrix on T is [Tr]m×m = [ti, tj]r. Set T = {t1, t2, …, tm} is said to be gcd-closed if (ti, tj) ∈ T for every ti and tj in T. In this paper, we give a generalization for the power GCD and LCM matrices defined on gcd-closed sets over unique factorization domains (UFDs). Moreover, we present a speculation for a generalization of Bourque–Ligh conjecture to UFDs which states that the least common multiple matrix defined on a gcd-closed P-ordered set in any UFD is nonsingular. Some examples that show what is done are additionally given in ℤ[i] and ℤp[x].
The identities for generalized Fibonacci numbers via orthogonal projection
Original research paper. Pages 167–177
Yasemin Alp and E. Gökçen Koçer
Full paper (PDF, 171 Kb) | Abstract
In this paper, we consider the space R(p, 1) of generalized Fibonacci sequences and orthogonal bases of this space. Using these orthogonal bases, we obtain the orthogonal projection onto a subspace R(p, 1) of ℝn. By using the orthogonal projection, we obtain the identities for the generalized Fibonacci numbers.
Relations for generalized Fibonacci and Tribonacci sequences
Original research paper. Pages 178–192
Robert Frontczak
Full paper (PDF, 184 Kb) | Abstract
In this article, we are concerned with connections between generalized Fibonacci and Tribonacci sequences. The identities we derive are of convolution type. As particular examples, we state several identities between Fibonacci and Tribonacci numbers, Fibonacci and Tribonacci–Lucas numbers, Lucas and Tribonacci numbers and Lucas and Tribonacci–Lucas numbers, respectively. Our results provide extensions of some recently obtained identities.
Generalized Fibonacci numbers and Bernoulli polynomials
Original research paper. Pages 193–198
Anthony G. Shannon, Ömür Deveci and Özgűr Erdağ
Full paper (PDF, 82 Kb) | Abstract
Relationships, in terms of equations and congruences, are developed between the Bernoulli numbers and arbitrary order generalizations of the ordinary Fibonacci and Lucas numbers. Some of these are direct connections and others are analogous similarities.
A note on the OEIS sequence A228059
Original research paper. Pages 199–205
Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada
Full paper (PDF, 153 Kb) | Abstract
The OEIS sequence A228059 lists odd numbers of the form p1+4kr2, where p is prime of the form 1+4m, r > 1, and gcd(p, r) = 1 that are closer to being perfect than previous terms. In this note, we present the prime factorizations of the first 37 terms.
Representation of higher even-dimensional rhotrix
Original research paper. Pages 206–219
A. O. Isere
Full paper (PDF, 187 Kb) | Abstract
The multiplication of higher even-dimensional rhotrices is presented and generalized. The concept of empty rhotrix, and the necessary and sufficient conditions for an even-dimensional rhotrix to be represented over a linear map, are investigated and presented.
Corrigendum to “Study of some equivalence classes of primes” [Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 2, 21–29]
Corrigendum. Page 220
Sadani Idir
Corrigendum (PDF, 78 Kb)
This issue of the International Journal “Notes on Number Theory and Discrete Mathematics” is published with the financial support of the Bulgarian National Science Fund, Grant Ref. No. KP-06-NP-28/2018.
Volume 25 ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4
