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 $$p_1+p_2+\cdots +p_k>(k-ϵ)p_t.$$

It is shown that Dickson’s Conjecture about primes in linear polynomials implies that if f is a reducible quadratic polynomial with integral coefficients and non-zero discriminant then for every r there exists an integer $N_r$ such that the polynomial $f\left(X\right)/N_r$ represents at least r distinct primes.

The aim of this paper is to prove the following Theorem Theorem Let $K$ be an octic subfield of the field $Q({\zeta }_p+{\zeta }_p^{-1})$ and let $p=n^4+16$ be prime. Then $p$ divides $h_K$ if and only if $p$ divides $B_j$ for some $j=\frac{p-1}{8}$, $3\frac{p-1}{8}$, $5\frac{p-1}{8}$, $7\frac{p-1}{8}$.

We are interested whether there is a nonnegative integer $u_0$ and an infinite sequence of digits $u_1,u_2,u_3,\cdots$ in base $b$ such that the numbers $u_0{b}^n+u_1{b}^{n-1}+\cdots +u_{n-1}b+u_n,$ where $n=0,1,2,\cdots ,$ are all prime or at least do not have prime divisors in a finite set of prime numbers $S.$ If any such sequence contains infinitely many elements divisible by at least one prime number $p\in S,$ then we call the set $S$ unavoidable with respect to $b$. It was proved earlier that unavoidable sets in base $b$ exist if $b\in \{2,3,4,6\},$ and that no unavoidable set exists in base $b=5.$ Now,...

Let $a_{d-1},\cdots ,a_0\in \mathbb{Z}$, where $d\in \mathbb{N}$ and $a_0\ne 0$, and let $X={\left(x_n\right)}_{n=1}^{\infty }$ be a sequence of integers given by the linear recurrence $x_{n+d}=a_{d-1}{x}_{n+d-1}+\cdots +a_0{x}_n$ for $n=1,2,3,\cdots$. We show that there are a prime number $p$ and $d$ integers $x_1,\cdots ,x_d$ such that no element of the sequence $X={\left(x_n\right)}_{n=1}^{\infty }$ defined by the above linear recurrence is divisible by $p$. Furthermore, for any nonnegative integer $s$ there is a prime number $p\ge 3$ and $d$ integers $x_1,\cdots ,x_d$ such that every element of the sequence $X={\left(x_n\right)}_{n=1}^{\infty }$ defined as above modulo $p$ belongs to the set $\{s+1,s+2,\cdots ,p-s-1\}$.

Improving on a theorem of Heath-Brown, we show that if X is sufficiently large then a positive proportion of the values n³ + 2: n ∈ (X,2X] have a prime factor larger than $X^{1+{10}^{-52}}$.

A classical theorem of M. Fried [2] asserts that if non-zero integers $\beta ₁,...,{\beta }_l$ have the property that for each prime number p there exists a quadratic residue ${\beta }_j$ mod p then a certain product of an odd number of them is a square. We provide generalizations for power residues of degree n in two cases: 1) n is a prime, 2) n is a power of an odd prime. The proofs involve some combinatorial properties of finite Abelian groups and arithmetic results of [3].