Volume 16, 2010, Number 3

Integer structure illustrates how primes represented by 4R₁ + 1 are equal to a sum of squares. Such primes are in Class ̅1₄ ⊂ Z₄, a modular ring. The rows of squares in Z₄ are well defined and this permits equalities for the primes to be derived from the integer structure. These equalities have the forms R₁ = r₁ + r₀ where r₁ = 3n(3n ± 1) or 2 + 9n′(n′ + 1) and r₀ = 2^2q(12m(3m ± 1) + 1) or 2^2q(4(2 + 9m′(m′ + 1)) + 1), with q = 0, 1, 2, 3,… and n, m yielding the pentagonal numbers, and n′, m′ the triangular numbers. When n = m the equations are similar to Euler’s prime equation. Equations for the remaining primes, in Class ̅3₄ may be obtained in the same manner using p = y² − x² with (y − x) = 1.

This paper considers some generalizations of the Fibonacci and Lucas numbers which are essentially ratios of the former, and hence not necessarily integers. Nevertheless, some new and elegant results emerge as well as variations on well-established identities.

In the paper the infinite power towers which are generated by an algebraic numbers belonging to the closed interval are investigated and an answer is given to the question when they are transcendental or rational numbers. Also a necessary condition for an infinite power tower to be an irrational algebraic number is proposed.

An interesting property of arithmetic functions φ, ψ and σ is being discussed and illustrated.