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It is conjectured that the sum $$ S_r(n)=\sum_{k=1}^{n} \frac{k}{k+r}\binom{n}{k} $$ for positive integers $r,n$ is never integral. This has been shown for $r\le 22$. In this note we study the problem in the ``$n$ aspect" showing that the…

Given $r \in \mathbb{N}$, define the function $S_{r}: \mathbb{N} \rightarrow \mathbb{Q}$ by $S_{r}(n)=\displaystyle \sum_{k=0}^{n} \frac{k}{k+r} \binom{n}{k}$. In $2015$, the second author conjectured that there are infinitely many $r \in…

We consider certain generalized binomial sums $\mathcal{S}_{(r,n)}(\ell)$ and discuss the nonintegrality of their values for integral parameters $n,r \geq 1$ and $\ell \in \mathbb{Z}$ in several cases using $p$-adic methods. In particular,…

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer…

For any integer $r \geq 1$, the sequence of numbers $\{{c^{(r)}_{k}}\}_{k \geq 0} $ is defined implicitly by [\sum_k\binom{n}{k}^r\binom{n+k}{k}^r = \sum_k\binom{n}{k}\binom{n+k}{k}c^{(r)}_k,\quad n=0,1,2,...] Asmus Schmidt conjectured that…

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…

For a fixed integer $k \ge 0$, consider representations of positive integers as sums of binomial coefficients of the form $\binom{n}{k}$. While exact minimal bounds for the number of required summands are known only in a few low-dimensional…

Let $S_k(m):=\sum_{j=1}^{m-1}j^k$ denote a power sum. In 2011, Kellner proposed the conjecture that for $m>3$ the ratio $S_k(m+1)/S_k(m)$ is never an integer, or, equivalently, that for any positive integer $a$, the equation $aS_k(m)=m^k$…

Let $n$ be a positive integer. In 1915, Theisinger proved that if $n\ge 2$, then the $n$-th harmonic sum $\sum_{k=1}^n\frac{1}{k}$ is not an integer. Let $a$ and $b$ be positive integers. In 1923, Nagell extended Theisinger's theorem by…

It is well known that the harmonic sum $H_n(1)=\sum_{k=1}^n\frac{1}{k}$ is never an integer for $n>1$. In 1946, Erd\H{o}s and Niven proved that the nested multiple harmonic sum $H_n(\{1\}^r)=\sum_{1\le k_1<\dots<k_r\le n}\frac{1}{k_1\cdots…

In this paper, we investigate the existence of Sierpi\'{n}ski numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer $r$, there exist infinitely many Sierpi\'{n}ski numbers and Riesel numbers of the…

In this paper we study a family of polynomials $$S_n^{(m)}(x):=\sum_{i,j=0}^n\binom ni^m\binom nj^m\binom{i+j}ix^{i+j}\ \ (m,n=0,1,2,\ldots).$$ For example, we show that $$\sum_{k=0}^{p-1}S_k^{(0)}(x)\equiv\frac…

Let $p$ be a prime. We define $S(p)$ the smallest number $k$ such that every positive integer is a sum of at most $k$ squares of integers that are not divisible by $p$. In this article, we prove that $S(2)=10$, $S(3)=6$, $S(5)=5$, and…

Let $a$ and $b$ be relatively prime positive integers. In this paper the weighted sum $\sum_{n\in{\rm NR}(a,b)}\lambda^{n-1}n^m$ is given explicitly or in terms of the Apostol-Bernoulli numbers, where $m$ is a nonnegative integer, and ${\rm…

For any integer $r\ge2$, define a sequence of numbers $\{c_k^{(r)}\}_{k=0}^\infty$, independent of the parameter $n$, by $$ \sum_{k=0}^n{\binom nk}^r{\binom{n+k}k}^r =\sum_{k=0}^n\binom nk\binom{n+k}kc_k^{(r)}, \qquad n=0,1,2,...c. $$ We…

Consider the equation $\mathcal{E}: x_1+ \cdots+x_{k-1} =x_{k}$ and let $k$ and $r$ be positive integers such that $r\mid k$. The number $S_{\mathfrak{z},2}(k;r)$ is defined to be the least positive integer $t$ such that for any 2-coloring…

Let $r$ and $k$ be positive integers with $r \mid k$. Denote by $S_{\mathrm{\mathfrak{z}}}(k;r)$ the minimum integer $n$ such that every coloring $\chi:[1,n] \rightarrow \{0,1,\dots,r-1\}$ admits a solution to $\sum_{i=1}^{k-1} x_i = x_k$…

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…

A partition of a positive integer $n$ is a representation of $n$ as a sum of a finite number of positive integers (called parts). A trapezoidal number is a positive integer that has a partition whose parts are a decreasing sequence of…

Let $a_1,a_2,\dots,a_k$ be positive integers with $\gcd(a_1,a_2,\dots,a_k)=1$. Frobenius number is the largest positive integer that is NOT representable in terms of $a_1,a_2,\dots,a_k$. When $k\ge 3$, there is no explicit formula in…