Volume 17, 2011, Number 1

Two modifications of Klamkin’s inequality are formulated and proved.

Using clear and straightforward approaches, we prove new ternary (base 3) digit extraction BBP-type formulas for polylogarithm constants. Some known results are also rediscovered in a more direct and elegant manner. An hitherto unproved degree 4 ternary formula is also proved. Finally, a couple of ternary zero relations are established, which prove two known but hitherto unproved formulas.

We introduce the concept of rhomtrees as a graphical method of representing rhotrices and present the relationships of their graphs with existing graphical models of some real world situations. These models include the topology of computing network, energy resource distribution network, methane compound and certain products of sets.

Let m be a positive integer, p be an odd prime, and ℤ_p^m = ℤ / (p^m) be the ring of integers modulo p^m. Let Q(x) = Q(x₁, x₂, …, x_n) be a nonsingular quadratic form with integer coefficients. In this paper we shall prove that any nonsingular quadratic form Q(x) over ℤ, Q(x) is equivalent to a diagonal quadratic form (modulo p^m).

Integer structure analysis in the Ring Z₃ shows that the right-end digit (RED) couples (1,4), (5,6) and (5,0) for x², y² in the primitive Pythagorean triple (pPt) in the equation z² = x² + y² do not lead to the primitive form of triple. The rows of x², y² with these REDs do not add to the required form for z². Since 3 ∤ z , the row of z² must follow the pentagonal numbers. Common factors for x, y are also inconsistent with pPt formation so that the (x², y²). RED (5,0) may be discarded directly.