Related papers: New results related to a conjecture of Manickam an…

We say that a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in \mathfrak{S}_n$ has a peak at index $i$ if $\pi_{i-1} < \pi_i > \pi_{i+1}$. Let $\mathcal{P}(\pi)$ denote the set of indices where $\pi$ has a peak. Given a set $S$ of positive…

We prove that for any nonnegative integers $n$ and $r$ the binomial sum $$ \sum_{k=-n}^n\binom{2n}{n-k}k^{2r} $$ is divisible by $2^{2n-\min\{\alpha(n),\alpha(r)\}}$, where $\alpha(n)$ denotes the number of 1's in the binary expansion of…

In 1845, Bertrand conjectured that twice any prime strictly exceeds the next prime. Tchebichef proved Bertrand's postulate in 1850. In 1934, Ishikawa proved a stronger result: the sum of any two consecutive primes strictly exceeds the next…

Let $n$ and $r$ be positive integers. Define the numbers $S_n^{(r)}$ by $S_n^{(r)}=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}{k}(2k+1)^r.$ In this paper we prove some conjectures of Guo and Liu which extend some conjectures of Z.-W. Sun…

For each positive integer $n$, function $f$, and point $x$, the 1998 conjecture by Ghinchev, Guerragio, and Rocca states that the existence of the $n$-th Peano derivative $f_{(n)}(x)$ is equivalent to the existence of all $n(n+1)/2$…

We show that every set $A$ of natural numbers with positive upper density can be shifted to contain the restricted sumset $\{b_1 + b_2 : b_1, b_2\in B \text{ and } b_1 \neq b_2 \}$ for some infinite set $B \subset A$.

We prove that any \(2\)-connected graph \(G\) on \(n\) vertices with minimum degree \(\delta(G) \ge \frac{n}{4}+2\) contains a \(2\)-connected subgraph of order \(k\) for every integer \(k\) with \(4 \le k \le n\). This improves a previous…

The number of $n \times n$ matrices whose entries are either -1, 0, or 1, whose row- and column- sums are all 1, and such that in every row and every column the non-zero entries alternate in sign, is proved to be $[1!4! >...…

Erd\"{o}s and Niven proved in 1946 that for any positive integers $m$ and $d$, there are at most finitely many integers $n$ for which at least one of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ are integers.…

A practical number is a positive integer $n$ such that all positive integers less than $n$ can be written as a sum of distinct divisors of $n$. Leonetti and Sanna proved that, as $x \to +\infty$, the central binomial coefficient…

It is well-known that for any distinct positive integers $k$ and $n$, the numbers $2^{2^k}+1$ and $2^{2^n}+1$ are relatively prime. In this paper we consider the situation when 1 is replaced by some positive integer $d>1$

In previous work of this author it was conjectured that the sum of power sums $p_\lambda,$ for partitions $\lambda$ ranging over an interval $[(1^n), \mu]$ in reverse lexicographic order, is Schur-positive. Here we investigate this…

The well-known $abc$-conjecture concerns triples $(a,b,c)$ of non-zero integers that are coprime and satisfy ${a+b+c=0}$. The strong $n$-conjecture is a generalisation to $n$ summands where integer solutions of the equation ${a_1 + \ldots +…

This paper is intended as a sequel to a paper arXiv:0803.2636 written by four of the coauthors here. In the paper, they proved a stronger form of the Erd\H{o}s-Mirksy conjecture which states that there are infinitely many positive integers…

We discuss conjectures related to the following two conjectures: (1) for each complex numbers x_1,...,x_n there exist rationals y_1,...,y_n \in [-2^{n-1},2^{n-1}] such that \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in…

In 1999 Lyngs{\o} and Pedersen proposed a conjecture stating that every binary circular word of length $n$ with equal number of zeros and ones has an antipalindromic linear subsequence of length at least $\frac{2}{3}n$. No progress over a…

Given a rational $a=p/q$ and $N$ nonnegative $d$-dimensional real vectors $u_1$, ..., $u_N$, we show that it is always possible to choose $(d-1)+\lceil (pN-d+1)/q\rceil$ of them such that their sum is (componentwise) at least…

The Motzkin numbers $M_n=\sum_{k=0}^n\binom n{2k}\binom{2k}k/(k+1)$ $(n=0,1,2,\ldots)$ and the central trinomial coefficients $T_n$ ($n=0,1,2,\ldots)$ given by the constant term of $(1+x+x^{-1})^n$, have many combinatorial interpretations.…

The Goldbach conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. This conjecture was first proposed by German mathematician Christian Goldbach in 1742 and, despite being obviously true,…

In 1971 Cusick proved that every real number $x\in[0,1]$ can be expressed as a sum of two continued fractions with no partial quotients equal to $1$. In other words, if we define a set $$ S(k):= \{ x\in[0,1] : a_n(x) \geq k \text{ for all }…