**Volume 18** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**A new proof of Lucas’ Theorem**

*Original research paper. Pages 1—6*

Alexandre Laugier and Manjil P. Saikia

Full paper (PDF, 171 Kb) | Abstract

**Pellian sequences and squares**

*Original research paper. Pages 7—10*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 33 Kb) | Abstract

_{4}. Integer Structure Analysis of this yields multiple-square equations exemplified by primitive Pythagorean triples, the Hoppenot equation and the equation for a sphere centred at the origin. The structure breaks down for higher powered triples so that solutions are blocked. However, Euler’s extension of Fermat’s Last Theorem does not work as the structure does permit multiple power equations such as

*a*

^{5}+

*b*

^{5}+

*c*

^{5}+

*d*

^{5}=

*e*

^{5}.

**On the polynomial and maximal solutions to a functional equation arising from multiplication of quantum integers**

*Original research paper. Pages 11—39*

Lan Nguyen

Full paper (PDF, 257 Kb) | Abstract

*P*for which there exist maximal solutions Γ

*to these functional equations with support bases*

_{P}*P*. We also give an explicit description of these maximal solutions.

**The common minimal equitable dominating signed graphs**

*Original research paper. Pages 40—46*

P. Siva Kota Reddy and U. K. Misra

Full paper (PDF, 150 Kb) | Abstract

*CMED*(Σ), where are

*CMED*(Σ) are complementary signed graph and common minimal equitable dominating signed graph of Σ respectively.

**On Zagier’s conjecture for L(E, 2): A number field example**

*Original research paper. Pages 47—53*

Jeffrey Stopple

Full paper (PDF, 164 Kb) | Abstract

*E*defined over a real quadratic field

*F*, the details of an example of Zagier’s conjecture. This relates

*L*(

*E*, 2) to values of the elliptic dilogarithm function at a divisor in the Jacobian of

*E*which arises from

*K*-theory.

**Solving algebraic equations with Integer Structure Analysis**

*Original research paper. Pages 54—60*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 100 Kb) | Abstract

_{5}.

**Generalized Hurwitz series**

*Original research paper. Pages 61—68*

A. G. Shannon

Full paper (PDF, 99 Kb) | Abstract

**Remark on the hollow triangular and quadratic numbers**

*Original research paper. Pages 69—70*

Krassimir Atanassov

Full paper (PDF, 133 Kb) | Abstract

*n*-gonal numbers is introduced and it is illustrated with the cases of triangular and quadratic numbers.

**A comment on a result of Virgolici**

*Original research paper. Pages 71—72*

Mickey Polasub

Full paper (PDF, 103 Kb) | Abstract

*+ 1009*

^{x}*=*

^{y}*p*.

^{z}**Volume 18**▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4