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A pair of Lucas sequences Uₙ = (αⁿ-βⁿ)/(α-β) and Vₙ = αⁿ + βⁿ is famously associated with each polynomial x² - Px + Q ∈ ℤ[x] with roots α and β. It is the purpose of this paper to show that when the root field of x² - Px + Q is either ℚ(i), or ℚ(ω), where , there are respectively two and four other second-order integral recurring sequences of characteristic polynomial x² - Px + Q that are of the same kinship as the U and V Lucas sequences. These are, when ℚ(α,β) = ℚ(i), the G and the H sequences...
Hasse showed the existence and computed the Dirichlet density of the set of primes for which the order of is odd; it is . Here we mimic successfully Hasse’s method to compute the density of monic irreducibles in for which the order of is odd. But on the way, we are also led to a new and elementary proof of these densities. More observations are made, and averages are considered, in particular, an average of the ’s as varies through all rational primes.
Romanoff (1934) showed that integers that are the sum of a prime and a power of 2 have positive lower asymptotic density in the positive integers. We adapt his method by showing more generally the existence of a positive lower asymptotic density for integers that are the sum of a prime and a term of a given nonconstant nondegenerate integral linear recurrence with separable characteristic polynomial.
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