Swisher [‘On the supercongruence conjectures of van Hamme’, Res. Math. Sci. 2 (2015), Article no. 18] and He [‘Supercongruences on truncated hypergeometric series’, Results Math. 72 (2017), 303–317] independently proved that Van Hamme’s (G.2) supercongruence holds modulo $p^4$ for any prime $p\equiv 1\pmod {4}$ . Swisher also obtained an extension of Van Hamme’s (G.2) supercongruence for $p\equiv 3 \pmod 4$ and $p>3$ . In this note, we give new one-parameter generalisations of Van Hamme’s (G.2) supercongruence modulo $p^3$ for any odd prime p. Our proof uses the method of ‘creative microscoping’ introduced by Guo and Zudilin [‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329–358].

For a simple bipartite graph G, we give an upper bound for the regularity of powers of the edge ideal $I(G)$ in terms of its vertex domination number. Consequently, we explicitly compute the regularity of powers of the edge ideal of a bipartite Kneser graph. Further, we compute the induced matching number of a bipartite Kneser graph.

We use a linear algebra interpretation of the action of Hecke operators on Drinfeld cusp forms to prove that when the dimension of the $\mathbb {C}_\infty $ -vector space $S_{k,m}(\mathrm {{GL}}_2(\mathbb {F}_q[t]))$ is one, the Hecke operator $\mathbf {T}_t$ is injective on $S_{k,m}(\mathrm {{GL}}_2(\mathbb {F}_q[t]))$ and $S_{k,m}(\Gamma _0(t))$ is a direct sum of oldforms and newforms.

Let n and k be positive integers with $n\ge k+1$ and let $\{a_i\}_{i=1}^n$ be a strictly increasing sequence of positive integers. Let $S_{n, k}:=\sum _{i=1}^{n-k} {1}/{\mathrm {lcm}(a_{i},a_{i+k})}$ . In 1978, Borwein [‘A sum of reciprocals of least common multiples’, Canad. Math. Bull. 20 (1978), 117–118] confirmed a conjecture of Erdős by showing that $S_{n,1}\le 1-{1}/{2^{n-1}}$ . Hong [‘A sharp upper bound for the sum of reciprocals of least common multiples’, Acta Math. Hungar. 160 (2020), 360–375] improved Borwein’s upper bound to $S_{n,1}\le {a_{1}}^{-1}(1-{1}/{2^{n-1}})$ and derived optimal upper bounds for $S_{n,2}$ and $S_{n,3}$ . In this paper, we present a sharp upper bound for $S_{n,4}$ and characterise the sequences $\{a_i\}_{i=1}^n$ for which the upper bound is attained.

We give a new q-analogue of the (A.2) supercongruence of Van Hamme. Our proof employs Andrews’ multiseries generalisation of Watson’s $_{8}\phi _{7}$ transformation, Andrews’ terminating q-analogue of Watson’s $_{3}F_{2}$ summation, a q-Watson-type summation due to Wei–Gong–Li and the creative microscoping method, developed by the author and Zudilin [‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329–358]. As a conclusion, we confirm a weaker form of Conjecture 4.5 by the author [‘Some generalizations of a supercongruence of van Hamme’, Integral Transforms Spec. Funct. 28 (2017), 888–899].

We consider the set of elements in a translation of the middle-third Cantor set which can be well approximated by algebraic numbers of bounded degree. A doubling dimensional result is given, which enables one to conclude an upper bound on the dimension of the set in question for a generic translation.

Let $0\leq \alpha \leq \infty $ , $0\leq a\leq b\leq \infty $ and $\psi $ be a positive function defined on $(0,\infty )$ . This paper is concerned with the growth of $L_{n}(x)$ , the largest digit of the first n terms in the Lüroth expansion of $x\in (0,1]$ . Under some suitable assumptions on the function $\psi $ , we completely determine the Hausdorff dimensions of the sets

$$\begin{align*}E_\psi(\alpha)=\bigg\{x\in(0,1]: \lim\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=\alpha\bigg\} \end{align*}$$

and

$$\begin{align*}E_\psi(a,b)=\bigg\{x\in(0,1]: \liminf\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=a, \limsup\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=b\bigg\}. \end{align*}$$

Let $a,b$ and n be positive integers and let $S=\{x_1, \ldots , x_n\}$ be a set of n distinct positive integers. For ${x\in S}$ , define $G_{S}(x)=\{d\in S: d<x, \,d\mid x \ \mathrm {and} \ (d\mid y\mid x, y\in S)\Rightarrow y\in \{d,x\}\}$ . Denote by $[S^a]$ the $n\times n$ matrix having the ath power of the least common multiple of $x_i$ and $x_j$ as its $(i,j)$ -entry. We show that the bth power matrix $[S^b]$ is divisible by the ath power matrix $[S^a]$ if $a\mid b$ and S is gcd closed (that is, $\gcd (x_i, x_j)\in S$ for all integers i and j with $1\le i, j\le n$ ) and $\max _{x\in S} \{|G_S (x)|\}=1$ . This confirms a conjecture of Shaofang Hong [‘Divisibility properties of power GCD matrices and power LCM matrices’, Linear Algebra Appl. 428 (2008), 1001–1008].

For a positive integer n, let $\mathcal T(n)$ denote the set of all integers greater than or equal to n. A sum of generalised m-gonal numbers g is called tight $\mathcal T(n)$ -universal if the set of all nonzero integers represented by g is equal to $\mathcal T(n)$ . We prove the existence of a minimal tight $\mathcal T(n)$ -universality criterion set for a sum of generalised m-gonal numbers for any pair $(m,n)$ . To achieve this, we introduce an algorithm giving all candidates for tight $\mathcal T(n)$ -universal sums of generalised m-gonal numbers. Furthermore, we provide some experimental results on the classification of tight $\mathcal T(n)$ -universal sums of generalised m-gonal numbers.

We study Ohno–Zagier type relations for multiple t-values and multiple t-star values. We represent the generating function of sums of multiple t-(star) values with fixed weight, depth and height in terms of the generalised hypergeometric function $\,_3F_2$ . As applications, we get a formula for the generating function of sums of multiple t-(star) values of maximal height and a weighted sum formula for sums of multiple t-(star) values with fixed weight and depth.

Let a, b, c be fixed coprime positive integers with $\min \{a,b,c\}>1$ . We discuss the conjecture that the equation $a^{x}+b^{y}=c^{z}$ has at most one positive integer solution $(x,y,z)$ with $\min \{x,y,z\}>1$ , which is far from solved. For any odd positive integer r with $r>1$ , let $f(r)=(-1)^{(r-1)/2}$ and $2^{g(r)}\,\|\, r-(-1)^{(r-1)/2}$ . We prove that if one of the following conditions is satisfied, then the conjecture is true: (i) $c=2$ ; (ii) a, b and c are distinct primes; (iii) $a=2$ and either $f(b)\ne f(c)$ , or $f(b)=f(c)$ and $g(b)\ne g(c)$ .

By analogy with the trace of an algebraic integer $\alpha $ with conjugates $\alpha _1=\alpha , \ldots , \alpha _d$ , we define the G-measure $ {\mathrm {G}} (\alpha )= \sum _{i=1}^d ( |\alpha _i| + 1/ | \alpha _i | )$ and the absolute ${\mathrm G}$ -measure ${\mathrm {g}}(\alpha )={\mathrm {G}}(\alpha )/d$ . We establish an analogue of the Schur–Siegel–Smyth trace problem for totally positive algebraic integers. Then we consider the case where $\alpha $ has all its conjugates in a sector $| \arg z | \leq \theta $ , $0 < \theta < 90^{\circ }$ . We compute the greatest lower bound $c(\theta )$ of the absolute G-measure of $\alpha $ , for $\alpha $ belonging to $11$ consecutive subintervals of $]0, 90 [$ . This phenomenon appears here for the first time, conforming to a conjecture of Rhin and Smyth on the nature of the function $c(\theta )$ . All computations are done by the method of explicit auxiliary functions.

A number of recent papers have estimated ratios of the partition function $p(n-j)/p(n)$ , which appear in many applications. Here, we prove an easy-to-use effective bound on these ratios. Using this, we then study the second shifted difference of partitions, $f(\,j,n) := p(n) -2p(n-j) +p(n-2j)$ , and give another easy-to-use estimate of $f(\,j,n)$ . As applications of these, we prove a shifted convexity property of $p(n)$ , as well as giving new estimates of the k-rank partition function $N_k(m,n)$ and non-k-ary partitions along with their differences.

Let $\alpha $ be a totally positive algebraic integer of degree d, with conjugates $\alpha _1=\alpha , \alpha _2, \ldots , \alpha _d$ . The absolute $S_k$ -measure of $\alpha $ is defined by $s_k(\alpha )= d^{-1} \sum _{i=1}^{d}\alpha _i^k$ . We compute the lower bounds $\upsilon _k$ of $s_k(\alpha )$ for each integer in the range $2\leq k \leq 15$ and give a conjecture on the results for integers $k>15$ . Then we derive the lower bounds of $s_k(\alpha )$ for all real numbers $k>2$ . Our computation is based on an improvement in the application of the LLL algorithm and analysis of the polynomials in the explicit auxiliary functions.

Let Q be a quiver of type $\tilde {A}_n$ . Let $\alpha =\alpha _1+\alpha _2+\cdots +\alpha _s$ be the canonical decomposition. For the polynomials $M_Q(\alpha ,q)$ that count the number of isoclasses of representations of Q over ${\mathbb F}_q$ with dimension vector $\alpha $ , we obtain a precise relation between the degree of $M_Q(\alpha ,q)$ and that of $\prod _{i=1}^{s} M_Q(\alpha _i,q)$ for an arbitrary dimension vector $\alpha $ .

The Euler–Mascheroni constant $\gamma =0.5772\ldots \!$ is the $K={\mathbb Q}$ example of an Euler–Kronecker constant $\gamma _K$ of a number field $K.$ In this note, we consider the size of the $\gamma _q=\gamma _{K_q}$ for cyclotomic fields $K_q:={\mathbb Q}(\zeta _q).$ Assuming the Elliott–Halberstam Conjecture (EH), we prove uniformly in Q that

$$ \begin{align*} \frac{1}{Q}\sum_{Q<q\le 2Q} |{\gamma_q - \log q}| = o(\log Q).\end{align*} $$

In other words, under EH, the $\gamma _q /\!\log q$ in these ranges converge to the one point distribution at $1$ . This theorem refines and extends a previous result of Ford, Luca and Moree for prime $q.$ The proof of this result is a straightforward modification of earlier work of Fouvry under the assumption of EH.

We study pencils of hypersurfaces over finite fields $\mathbb {F}_q$ such that each of the $q+1$ members defined over $\mathbb {F}_q$ is smooth.

We describe finite soluble nonnilpotent groups in which every minimal nonnilpotent subgroup is abnormal. We also show that if G is a nonsoluble finite group in which every minimal nonnilpotent subgroup is abnormal, then G is quasisimple and $Z(G)$ is cyclic of order $|Z(G)|\in \{1, 2, 3, 4\}$ .

In this note, we investigate some products of subgroups and vanishing conjugacy class sizes of finite groups. We prove some supersolubility criteria for groups with restrictions on the vanishing conjugacy class sizes of their subgroups.

We prove that a ring R is an $n \times n$ matrix ring (that is, $R \cong \mathbb {M}_n(S)$ for some ring S) if and only if there exists a (von Neumann) regular element x in R such that $l_R(x) = R{x^{n-1}}$ . As applications, we prove some new results, strengthen some known results and provide easier proofs of other results. For instance, we prove that if a ring R has elements x and y such that $x^n = 0$ , $Rx+Ry = R$ and $Ry \cap l_{R}(x^{n-1}) = 0$ , then R is an $n \times n$ matrix ring. This improves a result of Fuchs [‘A characterisation result for matrix rings’, Bull. Aust. Math. Soc. 43 (1991), 265–267] where it is proved assuming further that the element y is nilpotent of index two and $x+y$ is a unit. For an ideal I of a ring R, we prove that the ring $(\begin {smallmatrix} R & I \\ R & R \end {smallmatrix})$ is a $2 \times 2$ matrix ring if and only if $R/I$ is so.