Wyniki wyszukiwania

1

Primefree shifted Lucas sequences

100%

Jones L.

Acta Arithmetica

|

2015

|

tom 170

|

nr 3

287-298

EN

We say a sequence $𝓢 = (sₙ)_{n≥0}$ is primefree if |sₙ| is not prime for all n ≥ 0, and to rule out trivial situations, we require that no single prime divides all terms of 𝓢. In this article, we focus on the particular Lucas sequences of the first kind, ${𝓤}_a=(uₙ)_{n≥0}$, defined by u₀ = 0, u₁ = 1, and uₙ = au_{n-1} + u_{n-2} for n≥2, where a is a fixed integer. More precisely, we show that for any integer a, there exist infinitely many integers k such that both of the shifted sequences $𝓤_a ± k$ are simultaneously primefree. This result extends previous work of the author for the single shifted sequence $𝓤_a - k$ when a = 1 to all other values of a, and establishes a weaker form of a conjecture of Ismailescu and Shim. Moreover, we show that there are infinitely many values of k such that every term of both of the shifted sequences $𝓤_a ± k$ has at least two distinct prime factors.

2

Using Lucas sequences to generalize a theorem of Sierpiński

100%

Jones L.

Acta Arithmetica

|

2012

|

tom 152

|

nr 3

307-322

3

A class of irreducible polynomials

63%

Harrington J., Jones L.

Colloquium Mathematicum

|

2013

|

tom 132

|

nr 1

113-119

EN

Let $f(x) = xⁿ + k_{n-1}x^{n-1} + k_{n-2}x^{n-2} + ⋯ +k₁x + k₀ ∈ ℤ[x]$, where $3 ≤ k_{n-1} ≤ k_{n-2} ≤ ⋯ ≤ k₁ ≤ k₀ ≤ 2k_{n-1} - 3$. We show that f(x) and f(x²) are irreducible over ℚ. Moreover, the upper bound of $2k_{n-1} - 3$ on the coefficients of f(x) is the best possible in this situation.

Ograniczanie wyników

2 Acta Arithmetica

1 Colloquium Mathematicum

3 Jones L.

1 Harrington J.

1 2015

1 2013

1 2012

Primefree shifted Lucas sequences

Using Lucas sequences to generalize a theorem of Sierpiński

A class of irreducible polynomials