**Volume 26** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**In Memoriam: Prof. Varanasi Sitaramaiah**

*Editorial. Page 1*

Editorial (PDF, 79 Kb)

**Bi-unitary multiperfect numbers, IV(a)**

*Original research paper. Pages 2–32*

Pentti Haukkanen and Varanasi Sitaramaiah

Full paper (PDF, 320 Kb) | Abstract

Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. The present paper is Part IV(a) in a series of papers on even bi-unitary multiperfect numbers. In parts I, II and III we found all bi-unitary triperfect numbers of the form , where and is odd. There exist exactly ten such numbers. In this part we solve partly the case . We prove that if is a bi-unitary triperfect number of the form , where , then . We then confine ourselves to the case . We prove that in this case we have and further show that is the only bi-unitary triperfect number of this form.

**On the quantity m^{2} − p^{k} where p^{k}m^{2} is an odd perfect number**

*Original research paper. Pages 33–38*

Jose Arnaldo Bebita Dris and Immanuel Tobias San Diego

Full paper (PDF, 193 Kb) | Abstract

**Congruences involving alternating sums related to harmonic numbers and binomial coefficients**

*Original research paper. Pages 39–51*

Laid Elkhiri, Miloud Mihoubi and Abdellah Derbal

Full paper (PDF, 187 Kb) | Abstract

**On a sum involving the number of distinct prime factors function related to the integer part function**

*Original research paper. Pages 52–56*

Mihoub Bouderbala and Meselem Karras

Full paper (PDF, 146 Kb) | Abstract

**Objects generated by an arbitrary natural number**

*Original research paper. Pages 57–62*

Krassimir Atanassov

Full paper (PDF, 158 Kb) | Abstract

*(*

__Set__*n*), generated by an arbitrary natural number

*n*, is defined. Some arithmetic functions, defined over its elements are introduced. Some of the arithmetic, set-theoretical and algebraic properties of the new objects are studied.

**On Pythagorean triplet semigroups**

*Original research paper. Pages 63–67*

Antoine Mhanna

Full paper (PDF, 142 Kb) | Abstract

**On a translated sum over primitive sequences related to a conjecture of Erdős**

*Original research paper. Pages 68–73*

Nadir Rezzoug, Ilias Laib and Guenda Kenza

Full paper (PDF, 220 Kb) | Abstract

where denotes the set of prime numbers.

**A study on some identities involving ( s_{k}, t)-Jacobsthal numbers**

*Original research paper. Pages 74–79*

Serpil Halici and Mine Uysal

Full paper (PDF, 159 Kb) | Abstract

**The translated Whitney–Lah numbers: generalizations and q-analogues**

*Original research paper. Pages 80–92*

Mahid M. Mangontarum

Full paper (PDF, 203 Kb) | Abstract

*q*-analogues of the said formulas and identities by establishing similar properties for the translated

*q*-Whitney numbers.

**Explicit formula of a new class of q-Hermite-based Apostol-type polynomials and generalization**

*Original research paper. Pages 93–102*

Mouloud Goubi

Full paper (PDF, 215 Kb) | Abstract

*q*-Hermite-based Apostol-type polynomials introduced by Waseem A. Khan and Divesh Srivastava. We give their explicit formula and study a generalized class depending in any

*q*-analog generating function.

**A note on the Fermat quartic 34 x^{4} + y^{4} = z^{4}**

*Original research paper. Pages 103–105*

Gustaf Söderlund

Full paper (PDF, 129 Kb) | Abstract

**On a generalization of the Monkey and Coconuts Problem**

*Original research paper. Pages 106–112*

Amitabha Tripathi

Full paper (PDF, 189 Kb) | Abstract

share for each sailor in each version.

We find explicit solutions for both the original version and the variation in the general case of *n* sailors in which at each stage *r* coconuts are tossed to the monkey. Even more generally, we also investigate the two versions when the *n* sailors leave *r*_{1}, …, *r _{n}* coconuts to the monkey.

**On algorithms for computing the sums of powers of arithmetic progressions**

*Original research paper. Pages 113–121*

Peter J. Shiue, Shen C. Huang and Eric Jameson

Full paper (PDF, 164 Kb) | Abstract

**The Gauss product and Raabe’s integral for k-gamma functions**

*Original research paper. Pages 122–127*

József Sándor

Full paper (PDF, 239 Kb) | Abstract

*k*-gamma functions. The Sándor–Tóth short product formula [16] is also attended to these functions. An asymptotic formula and Raabe’s integral analogue are also considered.

**Derangement polynomials with a complex variable**

*Original research paper. Pages 128–135*

Abdelkader Benyattou

Full paper (PDF, 170 Kb) | Abstract

**Properties of hyperbolic generalized Pell numbers**

*Original research paper. Pages 136–153*

Yüksel Soykan and Melih Göcen

Full paper (PDF, 219 Kb) | Abstract

**Padovan sequence generalization – a study of matrix and generating function**

*Original research paper. Pages 154–163*

Renata Passos Machado Vieira, Francisco Regis Vieira Alves and Paula Maria Machado Cruz Catarino

Full paper (PDF, 183 Kb) | Abstract

**New Tribonacci recurrence relations and addition formulas**

*Original research paper. Pages 164–172*

Kunle Adegoke, Adenike Olatinwo and Winning Oyekanmi

Full paper (PDF, 133 Kb) | Abstract

**On the generalized Fibonacci and Lucas 2 ^{k}−ions**

*Original research paper. Pages 173–186*

Sure Köme and Hafize Kirik

Full paper (PDF, 288 Kb) | Abstract

^{k}−ions which are the generalizations of several quaternions, octonions and higher order dimensional algebras. We give the generating functions, the Binet formulas and well-known identities such as Catalan’s identity and Cassini’s identity for the modified generalized Fibonacci and Lucas 2

^{k}−ions.

**Dual bicomplex Horadam quaternions**

*Original research paper. Pages 187–205*

Kübra Gül

Full paper (PDF, 220 Kb) | Abstract

**A note on the coefficient array of a generalized Fibonacci polynomial**

*Original research paper. Pages 206–212*

A. G. Shannon and Ömür Deveci

Full paper (PDF, 118 Kb) | Abstract

**Significant role of the specific prime number p = 257 in the improvement of cryptosystems**

*Original research paper. Pages 213–222*

Hana Ali-Pacha, Naima Hadj-Said, Adda Ali-Pacha and Özen Özer

Full paper (PDF, 118 Kb) | Abstract

(a) cryptography, which is interested in the security of information.

(b) cryptanalysis, which seeks to attack it.

One have a starting set of 256 elements, we add a new element to this set to form a set of 257 elements. In this paper, we consider a finite field that contains 257 elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers modulo *p* when *p* is a prime number. For our case ℤ/*p*ℤ, *p* = 257. We apply it to affine ciphers and show that this cipher looks like a permutation cipher. The idea based on this result, is to use the affine ciphers with the modulo 257 (as an initial permutation) in any specific algorithm of ciphering. Besides, one finishes with the decryption affine with the modulo 257 like an inverse permutation. This is to significantly increase the security of the specific encryption algorithm and to lengthen the 16-bits encryption key.

*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-NP1-15/2019.*