**Volume 23** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**On 2-powerfully perfect numbers in three quadratic rings**

*Original research paper. Pages 1–11*

Colin Defant

Full paper (PDF, 208 Kb) | Abstract

*n*-powerfully perfect numbers in these rings. This definition serves to extend the concept of perfect numbers that have been defined and studied in the integers. We investigate the properties of 2-powerfully perfect numbers in the rings and the three imaginary quadratic rings with unique factorization in which 2 is not a prime.

**Conditions equivalent to the Descartes–Frenicle–Sorli Conjecture on odd perfect numbers**

*Original research paper. Pages 12–20*

Jose Arnaldo B. Dris

Full paper (PDF, 156 Kb) | Abstract

*k*= 1 if

*q*

^{k}

*n*

^{2}is an odd perfect number with Euler prime

*q*. In this note, we present some conditions equivalent to this conjecture.

**Study of some equivalence classes of primes**

*Original research paper. Pages 21–29*

Sadani Idir

Full paper (PDF, 236 Kb) | Abstract

**On solutions of the Diophantine equation 1/ x_{1}+2/x_{2}+3/x_{3}+…+k/x_{k}=1 when 2 ≤ x_{1} < x_{2} < x_{3} < … < x_{k} are integers and k = x_{1} **

*Original research paper. Pages 30–35*

Nechemia Burshtein

Full paper (PDF, 166 Kb) | Abstract

*x*

_{1}when

*x*

_{1}≤ 4, the equation has at least three solutions. For the particular values

*x*

_{1}= 2, 3, 4, all the solutions of the equation are determined and demonstrated.

**The primitive solutions to the Diophantine equation 2 X^{4} + Y^{4} =Z^{3}**

*Original research paper. Pages 36–44*

Gustav Söderlund

Full paper (PDF, 181 Kb) | Abstract

*x*,

*y*,

*z*) =(±5,±3, 11). The proofs involved are based solely on elementary methods with no use of computers and the elliptic curve machinery.

**The q-Lah numbers and the nth q-derivative of exp_{q}(1/n)**

*Original research paper. Pages 45–47*

Jacob Katriel

Full paper (PDF, 145 Kb) | Abstract

*q*-Lah numbers as well, i.e., they can be obtained by taking successive

*q*-derivatives of exp

_{q}(1/

*n*), where exp

_{q}(

*x*) is the

*q*-exponential.

**On two arithmetic functions**

*Original research paper. Pages 48–53*

József Sándor and Krassimir T. Atanassov

Full paper (PDF, 162 Kb) | Abstract

**Fast converging series for zeta numbers in terms of polynomial representations of Bernoulli numbers**

*Original research paper. Pages 54–80*

J. Braun, D. Romberger and H. J. Bentz

Full paper (PDF, 254 Kb) | Abstract

*B*

_{2n}as a function of

*B*

_{2n − 2}. Furthermore, we show that a direct computation of the Riemann zeta-function and their derivatives at

*k*∈ Z is possible in terms of these polynomial representation. As an explicit example, our polynomial Bernoulli number representation is applied to fast approximate computations of ζ(3), ζ(5) and ζ(7).

**Extensions of D’Aurizio’s trigonometric inequality**

*Original research paper. Pages 81–83*

József Sándor

Full paper (PDF, 139 Kb) | Abstract

**A conjecture on degrees of algebraic equations**

*Original research paper. Pages 84–90*

Simon Davis

Full paper (PDF, 175 Kb) | Abstract

*X*,

*Y*and

*Z*equal to 1, and specialization to prime factors of the product of integer values of the polynomials yields the inequality equivalent to the

*abc*conjecture.

**On the Pell p-circulant sequences**

*Original research paper. Pages 91–103*

Yeşim Aküzüm, Ömür Deveci, and A. G. Shannon

Full paper (PDF, 197 Kb) | Abstract

*p,i*)-sequence and then, we obtain miscellaneous properties of these sequences. Also, we consider the cyclic groups which are generated by the generating matrices and the auxiliary equations of the defined recurrence sequences and then, we study the orders of these groups. Furthermore, we extend the Pell p-circulant sequence to groups. Finally, we obtain the lengths of the periods of Pell p-circulant sequences in the semidihedral group

*SD*

_{2m}for

*m*≥ 4 as applications of the results obtained.

**Closed-form evaluations of Fibonacci–Lucas reciprocal sums with three factors**

*Original research paper. Pages 104–116*

Robert Frontczak

Full paper (PDF, 189 Kb) | Abstract

**On Legendre’s Conjecture**

*Original research paper. Pages 117–125*

A. G. Shannon and J. V. Leyendekkers

Full paper (PDF, 179 Kb) | Abstract

**The extension of some D(4)-pairs**

*Original research paper. Pages 126–135*

Alan Filipin

Full paper (PDF, 181 Kb) | Abstract

*D*(4)-triple {

*a*,

*b*,

*c*} with

*a*<

*b*<

*a*+57√

*a*has a unique extension to a quadruple with a larger element. This furthermore implies that

*D*(4)-pair {

*a*,

*b*} cannot be extended to a a quintuple if

*a*<

*b*<

*a*+57√

*a*.