Related papers: Almost resolvable $k$-cycle systems with $k\equiv …

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for $\alpha\in\mathbb{R}\backslash\mathbb{Q},\,\beta\in\mathbb{R}$ and $0<\theta<10/1561$, there…

For any integer $k\ge 1$, we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order $2^k$. The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class…

For two integers $k$ and $\ell$, an $(\ell \text{ mod }k)$-cycle means a cycle of length $m$ such that $m\equiv \ell\pmod{k}$. In 1977, Bollob\'{a}s proved a conjecture of Burr and Erd\H{o}s by showing that if $\ell$ is even or $k$ is odd,…

In this note, we classify all finite groups having exactly 6, 7 or 8 cyclic subgroups. This gives a partial answer to the open problem posed by Tarnauceanu (Amer. Math. Monthly, 122 (2015), 275-276). As a consequence of our results, we also…

The middle levels problem is to find a Hamilton cycle in the middle levels, M_{2k+1}, of the Hasse diagram of B_{2k+1} (the partially ordered set of subsets of a 2k+1-element set ordered by inclusion). Previously, the best result was that…

In this note, we prove that every non-complete $(k+1)$-critical graph contains cycles of all lengths modulo $k$, where $k=4,5$.

In this paper new $1$-rotational 2-Steiner systems for different admissible $v,k$ pairs are introduced. In particular, $1$-rotational unitals of order $4$ are enumerated.

We mainly introduce two new kinds of numbers given by $$R_n=\sum_{k=0}^n\binom nk\binom{n+k}k\frac1{2k-1}\quad\ (n=0,1,2,...)$$ and $$S_n=\sum_{k=0}^n\binom nk^2\binom{2k}k(2k+1)\quad\ (n=0,1,2,...).$$ We find that such numbers have many…

Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper, by using the work of Ishii and Deuring's theorem for elliptic curves with complex multiplication we solve some conjectures of Zhi-Wei Sun concerning…

Let $S,T$ be two numerical semigroups. We study when $S$ is one half of $T$, with $T$ almost symmetric. If we assume that the type of $T$, $t(T)$, is odd, then for any $S$ there exist infinitely many such $T$ and we prove that $1 \leq t(T)…

This paper contributes to the conjecture of R. Scott and R. Styer which asserts that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive…

Let $f(k)$ be the maximum possible chromatic number of a graph whose edge set can be partitioned into at most $k$ complete bipartite graphs. Alon, Saks, and Seymour conjectured that $f(k)=k+1$ for all $k$. While the conjecture was verified…

Using a recently introduced framework, we derive criteria for quantum k-separability, which are very easily computed. In the case k = 2, our criteria are equally strong to the best methods known so far, while in all other cases there are…

We consider the problem of counting the copies of a length-$k$ pattern $\sigma$ in a sequence $f \colon [n] \to \mathbb{R}$, where a copy is a subset of indices $i_1 < \ldots < i_k \in [n]$ such that $f(i_j) < f(i_\ell)$ if and only if…

We present some congruences modulo $p^{6-d}$ for sums of the type $\sum_{k=0}^{(p-3)/2}x^k{2k\choose k}/(2k+1)^d$, for $d=1,2,3$ where $p>5$ is a prime.

For each $k\geq 3$, Green proved an arithmetic $k$-cycle removal lemma for any abelian group $G$. The best known bounds relating the parameters in the lemma for general $G$ are of tower-type. For $k>3$, even in the case $G=\mathbb{F}_2^n$…

For any fixed $k\geq 2$, we prove that every sufficiently large integer can be expressed as the sum of a $k$th power of a prime and a number with at most $M(k)=6k$ prime factors. For sufficiently large $k$ we also show that one can take…

In this paper, we first obtain some analogues of a formula of Zagier (1995) and Stanley (2011). For instance, we prove that the number of pairs of $n$-cycles whose product has $k$ cycles and has $m$ given elements contained in distinct…

We prove bounds for the number of solutions to $$a_1 + \dots + a_k = a_1' + \dots + a_k'$$ over $N$-element sets of reals, which are sufficiently convex or near-convex. A near-convex set will be the image of a set with small additive…

We consider the problem of maximizing a monotone submodular function in a $k$-exchange system. These systems, introduced by Feldman et al., generalize the matroid k-parity problem in a wide class of matroids and capture many other…