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Acta Arithmetica

### Primefree shifted Lucas sequences

Acta Arithmetica

We say a sequence $={\left(sₙ\right)}_{n\ge 0}$ 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 . In this article, we focus on the particular Lucas sequences of the first kind, ${}_{a}={\left(uₙ\right)}_{n\ge 0}$, defined by u₀ = 0, u₁ = 1, and uₙ = aun-1 + un-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 ${}_{a}±k$ are simultaneously primefree. This result extends...

### Generalizing a theorem of Schur

Czechoslovak Mathematical Journal

In a letter written to Landau in 1935, Schur stated that for any integer $t>2$, there are primes ${p}_{1}<{p}_{2}<\cdots <{p}_{t}$ such that ${p}_{1}+{p}_{2}>{p}_{t}$. In this note, we use the Prime Number Theorem and extend Schur’s result to show that for any integers $t\ge k\ge 1$ and real $ϵ>0$, there exist primes ${p}_{1}<{p}_{2}<\cdots <{p}_{t}$ such that ${p}_{1}+{p}_{2}+\cdots +{p}_{k}>\left(k-ϵ\right){p}_{t}.$

### A class of irreducible polynomials

Colloquium Mathematicae

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

### Variations on a theme of Sierpiński.

Journal of Integer Sequences [electronic only]

Integers

### Appending digits to generate an infinite sequence of composite numbers.

Journal of Integer Sequences [electronic only]

### Sequences of reducible $\left\{0,1\right\}$-polynomials modulo a prime.

Journal of Integer Sequences [electronic only]

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