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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 ${}_a\pm k$ are simultaneously primefree. This result extends...

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 $$p_1+p_2+\cdots +p_k>(k-ϵ)p_t.$$

Let $f\left(x\right)=xⁿ+k_{n-1}{x}^{n-1}+k_{n-2}{x}^{n-2}+\cdots +k₁x+k₀\in \mathbb{Z}\left[x\right]$, where $3\le k_{n-1}\le k_{n-2}\le \cdots \le k₁\le k₀\le 2k_{n-1}-3$. 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.