相关论文: An improvement of Prouhet's 1851 result on multigr…

Sequence of positive integers $\{x_n\}_{n\geq1}$ is called similar to $\mathbb {N}$ respectively a given property $A$ if for every $n\geq1$ the numbers $x_n$ and $n$ are in the same class of equivalence respectively $A\enskip(x_n\sim n…

Gijswijt's sequence consists almost entirely of small positive integers. However, it is known that every positive integer eventually appears in the sequence. In this paper we determine its growth rate. Specifically, we prove that for…

In 1961, P. Erd\H{o}s, A. Ginzburg, and A. Ziv proved a remarkable theorem stating that each set of $2n-1$ integers contains a subset of size $n$, the sum of whose elements is divisible by $n$. We will prove a similar result for pairs of…

In this paper we study $N_d(k)$ the smallest positive integer such that any nice measure $\mu$ in $\R^d$ can be partitioned in $N_d(k)$ parts of equal measure so that every hyperplane avoids at least $k$ of them. A theorem of Yao and Yao…

For a fixed positive integer $m$ and any partition $m = m_1 + m_2 + \cdots + m_e$ , there exists a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that $$m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots+\frac{1}{n_{k}},$$ with the property…

Lately, Werner Schulte has conjectured that for all positive $n>1$, $n$ divides $\frac{(n-2)! (n-1)!}{2^{n-3}} + 4$ if and only if $n$ is prime. In this paper, We use elementary methods, to give a simple proof of this conjecture.

Making use of covering systems and a theorem of D. Shiu, the first and second authors showed that for every positive integer $k$, there exist $k$ consecutive widely digitally delicate primes. They also noted that for every positive integer…

For any positive integer r, the r-associated Stirling number of the second kind enumerates the number of partitions of the set{1,2,3,...,n} into k non-empty disjoint subsets such that each subset contains at least r elements. We introduce…

In this paper we obtained an original integer sequence based on the properties of the multinomial coefficient. We investigated a property of the sequence that shows connection with a primality testing. For any prime n the n-th term in the…

It is a classical fact that every $n$-element set of positive reals has at least $\binom{n+1}{2}+1$ distinct subset sums, with equality exactly for homogeneous arithmetic progressions (when $n\geq 4$). We establish stability versions of…

A natural number is a binary $k$'th power if its binary representation consists of $k$ consecutive identical blocks. We prove an analogue of Waring's theorem for sums of binary $k$'th powers. More precisely, we show that for each integer $k…

Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a…

A positive integer $n$ is said to be a practical number if every integer in $[1,n]$ can be represented as the sum of distinct divisors of $n$. In this article, we consider practical numbers of a given polynomial form. We give a necessary…

For a fixed integer $k$, we define a sequence $A_k=(a_k(n))_{n\geq0}$ and a corresponding sparse subsequence $S_k$ using the cardinality of the $n$-th symmetric power of the set $\{1,2,\ldots, k\}$. For $k\in\{2,\dots,8\}$, we find…

We prove an upper bound for the number of representations of a positive integer $N$ as the sum of four $k$-th powers of integers of size at most $B$, using a new version of the Determinant method developed by Heath-Brown, along with recent…

We introduce two new classes of integers. The first class consists of numbers $N$ for which there exists at least one nonnegative integer $A$, such that the sum of $A$ and the sum of digits of $N$, added to the reversal of the sum, gives…

Recently, Kargin et al. (arXiv:2008.00284 [math.NT]) obtained (among many other things) the following formula for the hyper-sums of powers of integers $S_k^{(m)}(n)$ \begin{equation*} S_k^{(m)}(n) = \frac{1}{m!} \sum_{i=0}^{m} (-1)^i…

Let $n_0(N,k)$ be the number of initial Fourier coefficients necessary to distinguish newforms of level $N$ and even weight $k$. We produce extensive data to support our conjecture that if $N$ is a fixed squarefree positive integer and $k$…

We give a new short proof of the most simple relation between consecutive power sums of the first m positive integers.

A triple of positive integers (d,h,m) is admissible if for any m given masses in R^d there exist h hyperplanes that cut each of these masses into 2^h equal pieces. We present an elementary reduction which combined with results by Ramos…