For an integer k ≥ 2, let $\left(F{ₙ}^{\left(k\right)}\right)ₙ$ be the k-Fibonacci sequence which starts with 0,..., 0,1 (k terms) and each term afterwards is the sum of the k preceding terms. This paper completes a previous work of Marques (2014) which investigated the spacing between terms of distinct k-Fibonacci sequences.

This paper gives further results about the distribution in the arithmetic progressions (modulo a product of two primes) of reducible quadratic polynomials $(an+b)(cn+d)$ in short intervals $n\in [x,x+x^{\vartheta }]$, where now $\vartheta \in (0,1]$. Here we use the Dispersion Method instead of the Large Sieve to get results beyond the classical level $\vartheta$, reaching $3\vartheta /2$ (thus improving also the level of the previous paper, i.e. $3\vartheta -3/2$), but our new results are different in structure. Then, we make a graphical comparison of the two methods.

Let $g_j\left(m\right),j=1,2$, be complex-valued multiplicative functions. In the paper the necessary and sufficient conditions are indicated for the convergence in some sense of probability measure$$\frac{1}{n}\text{card}\left\{0\le m\le n:(g_1\left(m\right),g_2\left(m\right))\in A\right\},A\in ℬ\left({ℂ}^2\right),$$as $n\to \infty$.

A positive integer $n$ is called a square-free number if it is not divisible by a perfect square except $1$. Let $p$ be an odd prime. For $n$ with $(n,p)=1$, the smallest positive integer $f$ such that $n^f\equiv 1(modp)$ is called the exponent of $n$ modulo $p$. If the exponent of $n$ modulo $p$ is $p-1$, then $n$ is called a primitive root mod $p$. Let $A\left(n\right)$ be the characteristic function of the square-free primitive roots modulo $p$. In this paper we study the distribution $$\sum _{n\le x}A\left(n\right)A(n+1),$$ and give an asymptotic formula by using properties of character sums.

Hawkins introduced a probabilistic version of Erathosthenes’ sieve and studied the associated sequence of random “primes” ${\left(p_k\right)}_{k\ge 1}$. Using various probabilistic techniques, many authors have obtained sharp results concerning these random “primes”, which are often in agreement with certain classical theorems or conjectures for prime numbers. In this paper, we prove that the number of integers $k\le n$ such that $p_{k+\alpha }-p_k=\alpha$ is almost surely equivalent to $n/log{\left(n\right)}^{\alpha }$, for a given fixed integer $\alpha \ge 1$. This is a particular case of a recent...