Related papers: Salajan's conjecture on discriminating terms in an…

It is known that for an arbitrary positive integer \(n\) the sequence \(S(x^n)=(1^n, 2^n, \ldots)\) is complete, meaning that every sufficiently large integer is a sum of distinct \(n\)th powers of positive integers. We prove that every…

We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…

We study the Ramanujan-prime-counting function along the lines of Ramanujan's original work on Bertrand's Postulate. We show that the number of Ramanujan primes between x and 2x tends to infinity with x. This analysis leads us to define a…

Consider the sequence $\mathcal{V}(2,n)$ constructed in a greedy fashion by setting $a_1 = 2$, $a_2 = n$ and defining $a_{m+1}$ as the smallest integer larger than $a_m$ that can be written as the sum of two (not necessarily distinct)…

In 2016, Dannan and Sitnik established the notable Damascus inequality, which features a symmetric structure under a multiplicative constraint. In this study, we consider the natural generalisation of this inequality by characterising all…

The deletion distance between two binary words $u,v \in \{0,1\}^n$ is the smallest $k$ such that $u$ and $v$ share a common subsequence of length $n-k$. A set $C$ of binary words of length $n$ is called a $k$-deletion code if every pair of…

For two positive integers $m$ and $M$, we study the Davenport constant of the interval of integers $[\![ -m,M ]\!]$, that is the maximal length of a minimal zero-sum sequence composed of elements from $[\![ -m,M ]\!]$. We prove the…

We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let $n>3$ and let $c,d\in\Z$. If $n$ is composite, then \[ \det\big[(i^2+cij+dj^2)^{n-2}\big]_{0\leq i,j\leq n-1}\equiv 0\pmod {n^2}…

We define the second discriminant $D_2$ of a univariate polynomial $f$ of degree greater than $2$ as the product of the linear forms $2\,r_k-r_i-r_j$ for all triples of roots $r_i, r_k, r_j$ of $f$ with $i<j$ and $j\neq k, k\neq i$. $D_2$…

A digit $\pi_j$ in a permutation $\pi=[\pi_1,\ldots,\pi_n]\in S_n$ is defined to be a separator of $\pi$ if by omitting it from $\pi$ we get a new $2-$block. In this work we introduce a new statistic, the number of separators, on the…

We give a new interpretation of the derangement numbers d_n as the sum of the values of the largest fixed points of all non-derangements of length n-1. We also show that the analogous sum for the smallest fixed points equals the number of…

A set $\{a_i\:|\: 1\leq i \leq k\}$ of non-negative integers is a Golomb ruler if differences $a_i-a_j$, for any $i \neq j$, are all distinct.All finite Sidon sets are Golomb rulers, and vice versa. A set of $I$ disjoint Golomb rulers (DGR)…

Let $\left\{a_1, \dots, a_n\right\} \subset \mathbb{N}$ be a set of positive integers, $a_n$ denoting the largest element, so that for any two of the $2^n$ subsets the sum of all elements is distinct. Erd\H{o}s asked whether this implies…

The Collatz conjecture can be stated in terms of the reduced Collatz function R(x) = (3x+1)/2^m (where 2^m is the larger power of 2 that divides 3x+1). The conjecture is: Starting from any odd positive integer and repeating R(x) we…

The exponential local-global principle, or Skolem conjecture, says: Suppose that \(b\) is a positive integer, and that the sequence \((u_{n})_{n = -\infty}^{\infty}\) is such that every term is in \(\mathbb{Z}[1/b]\), the linear recurrence…

For positive integers $s$ and $L \geq 3$, Berkovich and Uncu (Ann. Comb. $23$ ($2019$) $263$--$284$) conjectured an inequality between the sizes of two closely related sets of partitions whose parts lie in the interval $\{s, \ldots, L+s\}$.…

For a fixed unit vector $a=(a_1,a_2,\ldots,a_n)\in S^{n-1}$, we consider the $2^n$ sign vectors $\varepsilon=(\varepsilon^1,\varepsilon^2,\ldots,\varepsilon^n)\in \{+1,-1\}^n$ and the corresponding scalar products $\varepsilon\cdot…

We study modular forms for the minimal index noncongruence subgroups of the modular group. Our main theorem is a proof of the unbounded denominator conjecture for these groups, and we also provide a study of the Fourier coefficients of…

Based on the Goldbach conjecture and arithmetic fundamental theorem, the Goldbach conjecture was extended to more general situations, i.e., any positive integer can be written as summation of some specific prime numbers, which depends on…

In this paper, we give a new upper bound for the number $N_{\mathcal{R}}$ which is defined to be the smallest positive integer such that a certain inequality due to Ramanujan involving the prime counting function $\pi(x)$ holds for every $x…