Proof of a recent conjecture of Z.-W. Sun

The polynomials $d_n(x)$ are defined by \begin{align*} d_n(x) &= \sum_{k=0}^n{n\choose k}{x\choose k}2^k. \end{align*} We prove that, for any prime $p$, the following congruences hold modulo $p$: \begin{align*} \sum_{k=0}^{p-1}\frac{{2k\choose k}}{4^k} d_k\left(-\frac{1}{4}\right)^2 &\equiv \begin{cases} 2(-1)^{\frac{p-1}{4}}x,&\text{if $p=x^2+y^2$ with $x\equiv 1\pmod{4}$,} 0,&\text{if $p\equiv 3\pmod{4}$,} \end{cases} [5pt] \sum_{k=0}^{p-1}\frac{{2k\choose k}}{4^k} d_k\left(-\frac{1}{6}\right)^2 &\equiv 0, \quad\text{if $p>3$,} [5pt] \sum_{k=0}^{p-1}\frac{{2k\choose k}}{4^k} d_k\left(\frac{1}{4}\right)^2 &\equiv \begin{cases} 0,&\text{if $p\equiv 1\pmod{4}$,} (-1)^{\frac{p+1}{4}}{\frac{p-1}{2}\choose \frac{p-3}{4}},&\text{if $p\equiv 3\pmod{4}$.} \end{cases} \sum_{k=0}^{p-1}\frac{{2k\choose k}}{4^k} d_k\left(\frac{1}{6}\right)^2 &\equiv 0, \quad\text{if $p>5$.} \end{align*} The $p\equiv 3\pmod{4}$ case of the first one confirms a conjecture of Z.-W. Sun, while the second one confirms a special case of another conjecture of Z.-W. Sun.Comment: 4 page

A congruence involving alternating harmonic sums modulo pαqβp^{\alpha}q^{\beta}

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \begin{equation*} \sum\limits_{i+j+k=p^{r}\atop{i,j,k\in \mathcal{P}_{p}}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} ~(\bmod ~ p^{r}), \end{equation*} where $\mathcal{P}_{n}$ denote the set of positive integers which are prime to $n$. In this note, we obtain the congruences for distinct odd primes $p,~q$ and positive integers $\alpha,~\beta$, \begin{equation*} \sum\limits_{i+j+k=p^{\alpha}q^{\beta}\atop{i,j,k\in\mathcal{P}_{pq}\atop{i\equiv j\equiv k\equiv 1\pmod{2}}}}\frac{1}{ijk}\equiv\frac{7}{8}(2-q)(1-\frac{1}{q^{3}})p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{\alpha}} \end{equation*} and \begin{equation*} \sum\limits_{i+j+k=p^{\alpha}q^{\beta}\atop{i,j,k\in \mathcal{P}_{pq}}}\frac{(-1)^{i}}{ijk} \equiv \frac{1}{2}(q-2)(1-\frac{1}{q^{3}})p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{\alpha}}. \end{equation*} Finally, we raise a conjecture that for $n>1$ and odd prime power $p^{\alpha}||n$, $\alpha\geq1$, \begin{eqnarray} \nonumber \sum\limits_{i+j+k=n\atop{i,j,k\in\mathcal{P}_{n}}}\frac{(-1)^{i}}{ijk} \equiv \prod\limits_{q|n\atop{q\neq p}}(1-\frac{2}{q})(1-\frac{1}{q^{3}})\frac{n}{2p}B_{p-3}\pmod{p^{\alpha}} \end{eqnarray} and \begin{eqnarray} \nonumber \sum\limits_{i+j+k=n\atop{i,j,k\in\mathcal{P}_{n}\atop{i\equiv j\equiv k\equiv 1\pmod{2}}}}\frac{1}{ijk} \equiv \prod\limits_{q|n\atop{q\neq p}}(1-\frac{2}{q})(1-\frac{1}{q^{3}})(-\frac{7n}{8p})B_{p-3}\pmod{p^{\alpha}}. \end{eqnarray

Equitable vertex arboricity of graphs

An equitable $(t,k,d)$-tree-coloring of a graph $G$ is a coloring to vertices of $G$ such that the sizes of any two color classes differ by at most one and the subgraph induced by each color class is a forest of maximum degree at most $k$ and diameter at most $d$. The minimum $t$ such that $G$ has an equitable $(t',k,d)$-tree-coloring for every $t'\geq t$ is called the strong equitable $(k,d)$-vertex-arboricity and denoted by $va^{\equiv}_{k,d}(G)$. In this paper, we give sharp upper bounds for $va^{\equiv}_{1,1}(K_{n,n})$ and $va^{\equiv}_{k,\infty}(K_{n,n})$ by showing that $va^{\equiv}_{1,1}(K_{n,n})=O(n)$ and va^{\equiv}_{k,\infty}(K_{n,n})=O(n^{\1/2}) for every $k\geq 2$. It is also proved that $va^{\equiv}_{\infty,\infty}(G)\leq 3$ for every planar graph $G$ with girth at least 5 and $va^{\equiv}_{\infty,\infty}(G)\leq 2$ for every planar graph $G$ with girth at least 6 and for every outerplanar graph. We conjecture that $va^{\equiv}_{\infty,\infty}(G)=O(1)$ for every planar graph and $va^{\equiv}_{\infty,\infty}(G)\leq \lceil\frac{\Delta(G)+1}{2}\rceil$ for every graph $G$

Some Extensions to Touchard's Theorem on Odd Perfect Numbers

The multiplicative structure of an odd perfect number $n$, if any, is $n=\pi^\alpha M^2$, where $\pi$ is prime, $\gcd(\pi,M)=1$ and $\pi\equiv \alpha\equiv1\pmod{4}$. An additive structure of $n$, established by Touchard, is that "$\bigl(n\equiv 9\pmod{36}\bigr )$ OR $\bigl (n\equiv1\pmod{12}\bigr )$". A first extension of Touchard's result is that the proposition "$\bigl(n\equiv x^2\pmod{4 x^2}\bigr )$ OR $\bigl (n\equiv \pi\equiv1\pmod{4 x}\bigr )$" holds for $x=3$ (the extension is due to the fact that the second congruence contains also $\pi$). We further extend the proof to $x=\alpha+2$, $\alpha+2$ prime, with the restriction that the congruence modulo $4 x$ does not include $n$. Besides, we note that the first extension of Touchard's result holds also with an exclusive disjunction, so that $\pi\equiv 1\pmod{12}$ is a sufficient condition because $3\nmid n$.Comment: 5 page

On the equation x2−Dy2=nx^2-Dy^2=n

We propose a method to determine the solvability of the diophantine equation $x^2-Dy^2=n$ for the following two cases: $(1)$ $D=pq$, where $p,q\equiv 1 \mod 4$ are distinct primes with $(\frac{q}{p})=1$ and $(\frac{p}{q})_4(\frac{q}{p})_4=-1$. $(2)$ $D=2p_1p_2... p_m$, where $p_i\equiv 1 \mod 8,1\leq i\leq m$ are distinct primes and $D=r^2+s^2$ with $r,s \equiv \pm 3 \mod 8$.Comment: 19p

New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions

On a $3$D manifold, a Weyl geometry consists of pairs $(g, A) =$ (metric, $1$-form) modulo gauge $\widehat{g} = {\rm e}^{2\varphi} g$, $\widehat{A} = A + {\rm d}\varphi$. In 1943, Cartan showed that every solution to the Einstein-Weyl equations $R_{(\mu\nu)} - \frac{1}{3} R g_{\mu\nu} = 0$ comes from an appropriate $3$D leaf space quotient of a $7$D connection bundle associated with a 3$^{\rm rd}$ order ODE $y''' = H(x,y,y',y'')$ modulo point transformations, provided $2$ among $3$ primary point invariants vanish $$\text{W\"unschmann}(H) \equiv 0\equiv \text{Cartan}(H).$$We find that point equivalence of a single PDE $z_y = F(x,y,z,z_x)$ with para-CR integrability $DF := F_x + z_x F_z \equiv 0$ leads to a completely similar $7$D Cartan bundle and connection. Then magically, the (complicated) equation $\text{W\"unschmann}(H) \equiv 0$ becomes $$0\equiv\text{Monge}(F):=9F_{pp}^2F_{ppppp}-45F_{pp}F_{ppp}F_{pppp}+40F_{ppp}^3,\qquad p:=z_x,$$ whose solutions are just conics in the $\{p, F\}$-plane. As an ansatz, we take $$F(x,y,z,p):= \frac{\alpha(y)(z-xp)^2+\beta(y)(z-xp)p+\gamma(y)(z-xp) +\delta(y)p^2+\varepsilon(y)p+\zeta(y)}{\lambda(y)(z-xp)+\mu(y) p+\nu(y)},$$ with $9$ arbitrary functions $\alpha, \dots, \nu$ of $y$. This $F$ satisfies $DF \equiv 0 \equiv \text{Monge}(F)$, and we show that the condition $\text{Cartan}(H) \equiv 0$ passes to a certain $\boldsymbol{K}(F) \equiv 0$ which holds for any choice of $\alpha(y), \dots, \nu(y)$. Descending to the leaf space quotient, we gain $\infty$-dimensional functionally parametrized and explicit families of Einstein-Weyl structures $\big[ (g, A) \big]$ in $3$D. These structures are nontrivial in the sense that ${\rm d}A \not\equiv 0$ and $\text{Cotton}([g]) \not \equiv 0$

Ramanujan-type Congruences for Overpartitions Modulo 16

Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Recently, Fortin-Jacob-Mathieu and Hirschhorn-Sellers independently obtained 2-, 3- and 4-dissections of the generating function for $\overline{p}(n)$ and derived a number of congruences for $\overline{p}(n)$ modulo $4$, $8$ and $64$ including $\overline{p}(5n+2)\equiv 0 \pmod{4}$, $\overline{p}(4n+3)\equiv 0 \pmod{8}$ and $\overline{p}(8n+7)\equiv 0 \pmod{64}$. By employing dissection techniques, Yao and Xia obtained congruences for $\overline{p}(n)$ modulo $8, 16$ and $32$, such as $\overline{p}(48n+26) \equiv 0 \pmod{8}$, $\overline{p}(24n+17)\equiv 0 \pmod{16}$ and $\overline{p}(72n+69)\equiv 0 \pmod{32}$. In this paper, we give a 16-dissection of the generating function for $\overline{p}(n)$ modulo 16 and we show that $\overline{p}(16n+14)\equiv0\pmod{16}$ for $n\ge 0$. Moreover, by using the $2$-adic expansion of the generating function of $\overline{p}(n)$ due to Mahlburg, we obtain that $\overline{p}(\ell^2n+r\ell)\equiv0\pmod{16}$, where $n\ge 0$, $\ell \equiv -1\pmod{8}$ is an odd prime and $r$ is a positive integer with $\ell \nmid r$. In particular, for $\ell=7$, we get $\overline{p}(49n+7)\equiv0\pmod{16}$ and $\overline{p}(49n+14)\equiv0\pmod{16}$ for $n\geq 0$. We also find four congruence relations: $\overline{p}(4n)\equiv(-1)^n\overline{p}(n) \pmod{16}$ for $n\ge 0$, $\overline{p}(4n)\equiv(-1)^n\overline{p}(n)\pmod{32}$ for $n$ being not a square of an odd positive integer, $\overline{p}(4n)\equiv(-1)^n\overline{p}(n)\pmod{64}$ for $n\not\equiv 1,2,5\pmod{8}$ and $\overline{p}(4n)\equiv(-1)^n\overline{p}(n)\pmod{128}$ for $n\equiv 0\pmod{4}$.Comment: 12 page

Long Cycles in 1-tough Graphs

In 1952, Dirac proved that every 2-connected graph with minimum degree $\delta$ either is hamiltonian or contains a cycle of length at least $2\delta$. In 1986, Bauer and Schmeichel enlarged the bound $2\delta$ to $2\delta+2$ under additional 1-tough condition - an alternative and more natural necessary condition for a graph to be hamiltonian. In fact, the bound $2\delta+2$ is sharp for a graph on $n$ vertices when $n\equiv 1(mod\ 3)$. In this paper we present the final version of this result which is sharp for each $n$: every 1-tough graph either is hamiltonian or contains a cycle of length at least $2\delta+2$ when $n\equiv 1(mod\ 3)$, at least $2\delta+3$ when $n\equiv 2(mod\ 3)$ or $n\equiv 1(mod\ 4)$, and at least $2\delta+4$ otherwise.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1204.651

A simple polynomial for a simple transposition

In this note, we review some facts about polynomials representing functions modulo primes p. In addition we prove that the polynomial f(x) = x^{p-2} + x^{p-3} + ... + x^3 + x^2 + 2x + 1 represents the transposition (0 1) modulo p, that is, f(0) \equiv 1 (mod p), f(1) \equiv 0 (mod p), and f(a) \equiv a (mod p) for all 2 \le a \le p-1.Comment: 4 page

Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum

The numbers $R_n$ and $W_n$ are defined as \begin{align*} R_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{1}{2k-1},\ \text{and}\ W_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{3}{2k-3}. \end{align*} We prove that, for any positive integer $n$ and odd prime $p$, there hold \begin{align*} \sum_{k=0}^{n-1}(2k+1)R_k^2 &\equiv 0 \pmod{n}, \\ \sum_{k=0}^{p-1}(2k+1)R_k^2 &\equiv 4p(-1)^{\frac{p-1}{2}} -p^2 \pmod{p^3}, \\ 9\sum_{k=0}^{n-1}(2k+1)W_k^2 &\equiv 0 \pmod{n}, \\ \sum_{k=0}^{p-1}(2k+1)W_k^2 &\equiv 12p(-1)^{\frac{p-1}{2}}-17p^2 \pmod{p^3}, \quad\text{if $p>3$.} \end{align*} The first two congruences were originally conjectured by Z.-W. Sun. Our proof is based on the multi-variable Zeilberger algorithm and the following observation: $${2n\choose n}{n\choose k}{m\choose k}{k\choose m-n}\equiv 0\pmod{{2k\choose k}{2m-2k\choose m-k}},$$ where $0\leqslant k\leqslant n\leqslant m \leqslant 2n$.Comment: 18 page