Related papers: On prime factors of Mersenne numbers

For $0\leq \alpha<1$ and prime number $p$ let $L(\alpha,p)$ be the sum of the first $[\alpha p]$ values of Legendre symbol modulo $p$. We study positivity of $L(\alpha,p)$ and prove that for $|\alpha-\frac13|<2\cdot 10^{-6}$ and for…

Denote $f(n):=\sum_{1\le k\le n} \tau(2^k-1)$, where $\tau$ is the number of divisors function. Motivated by a question of Paul Erd\H{o}s, we show that the sequence of ratios $f(2n)/f(n)$ is unbounded. We also present conditional results on…

Let $m$ be any positive integer and let $\delta_1,\delta_2\in\{1,-1\}$. We show that for some constanst $C_m>0$ there are infinitely many integers $n>1$ with $p_{n+m}-p_n\le C_m$ such that $$\left(\frac{p_{n+i}}{p_{n+j}}\right)=\delta_1\…

The non-associative exponential series $exp(x)$ is a power series with monomials from the magma $M$ of finite, planar rooted trees. The coefficient $a(t)$ of $exp(x)$ relative to a tree $t$ of degree $n$ is a rational number and it is shown…

Let $1<c<37/18,\,c\neq2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in(N,2N],$ the Diophantine inequality $|p_1^c+p_2^c+p_3^c-R|<\log^{-1}N$ is solvable in primes $p_1,\,p_2,\,p_3.$…

In this paper we introduce a new diophantine inequality with prime numbers. Let $1<c<\frac{10}{9}$. We show that for any fixed $\theta>1$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the tangent…

We determine all triples $(a,b,n)$ of positive integers such that $a$ and $b$ are relatively prime and $n^k$ divides $a^n + b^n$ (respectively, $a^n - b^n$), when $k$ is the maximum of $a$ and $b$ (in fact, we answer a slightly more general…

Given integer $n > 0$ and $m > 1$, we call a partition of set $[n] = \{1, \dots, n\}$ {\em $m$-good} if each of the partitioning sets is of size at most $m$ and the sum of numbers in it is a power of $m$, that is, $m^t$ for some $t \geq 0$.…

In this paper we construct a cover {a_s(mod n_s)}_{s=1}^k of Z with odd moduli such that there are distinct primes p_1,...,p_k dividing 2^{n_1}-1,...,2^{n_k}-1 respectively. Using this cover we show that for any positive integer m divisible…

The Mertens function, $M(x) := \sum_{n \leq x} \mu(n)$, is defined as the summatory function of the classical M\"obius function. The Dirichlet inverse function $g(n) := (\omega+1)^{-1}(n)$ is defined in terms of the shifted strongly…

Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding…

Given a partition $\{E_0,\ldots,E_n\}$ of the set of primes and a vector $\mathbf{k} \in \mathbb{N}_0^{n+1}$, we compute an asymptotic formula for the quantity $|\{m \leq x: \omega_{E_j}(m) = k_j \ \forall \ 0 \leq j \leq n\}|$ uniformly in…

We will prove that for every $m\geq 0$ there exists an $\varepsilon=\varepsilon(m)>0$ such that if $0<\lambda<\varepsilon$ and $x$ is sufficiently large in terms of $m$ and $\lambda$, then $$|\lbrace n\leq x: |[n,n+\lambda\log n]\cap…

We present the first fixed-length elementary closed-form expressions for the prime-counting function, $\pi(n)$, and the $n$-th prime number, $p(n)$. These expressions are arithmetic terms, requiring only a finite and fixed number of…

Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $k\ge 2$ whose first $k$ terms are $0,\ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. For an integer $m$,…

Let $M$ be a fixed positive integer. Let $(R_{j}(n))_{n\ge 1}$ be a linear recurrence sequence for every $j=0,1,\ldots, M$, and we set $f(n)=(R_0\circ \cdots \circ R_M)(n)$, where $(S\circ T)(n)= S(T(n))$. In this paper, we obtain…

We study some divisibility properties of multiperfect numbers. Our main result is: if $N=p_1^{\alpha_1}... p_s^{\alpha_s} q_1^{2\beta_1}... q_t^{2\beta_t}$ with $\beta_1, ..., \beta_t$ in some finite set S satisfies…

Let $P(m)$ denote the largest prime factor of an integer $m\geq 2$, and put $P(0)=P(1)=1$. For an integer $k\geq 2$, let $(F_{n}^{(k)})_{n\geq 2-k}$ be the $k-$generalized Fibonacci sequence which starts with $0,...,0,1$ ($k$ terms) and…

Let $a$ and $m>0$ be integers. We show that for any integer $b$ relatively prime to $m$, the set $\{a^n+bn:\ n=1,\ldots,m^2\}$ contains a complete system of residues modulo $m$. We also pose several conjectures for further research; for…

We describe a primality test for number $M=(2p)^{2^n}+1$ with odd prime $p$ and positive integer $n$. And we also give the special primality criteria for all odd primes $p$ not exceeding 19. All these primality tests run in polynomial time…