On some congruences of certain binomial sums

For any prime \(p>3,\) we prove that

$$\begin{aligned} \sum _{k=0}^{p-1}\frac{3k+1}{(-8)^k}{2k\atopwithdelims ()k}^3\equiv p\left( \frac{-1}{p}\right) +p^3E_{p-3}\pmod {p^4}, \end{aligned}$$

where \(E_{0},E_{1},E_{2},\ldots \) are Euler numbers and \(\left( \frac{\cdot }{p}\right) \) is the Legendre symbol. This result confirms a conjecture of Z.-W. Sun. We also re-prove that for any odd prime \(p,\)

$$\begin{aligned} \sum _{k=0}^{\frac{p-1}{2}}\frac{6k+1}{(-512)^k}{2k\atopwithdelims ()k}^3\equiv p\left( \frac{-2}{p}\right) \pmod {p^2} \end{aligned}$$

using WZ method.

     

On sums of binomial coefficients involving Catalan and Delannoy numbers modulo $$p^2$$

In this paper, we prove some congruences conjectured by Z.-W. Sun: For any prime \(p>3\), we determine

$$\begin{aligned} \sum \limits _{k = 0}^{p - 1} {\frac{{{C_k}C_k^{(2)}}}{{{{27}^k}}}} \quad {\text { and }}\quad \sum \limits _{k = 1}^{p - 1} {\frac{{\left( {\begin{array}{l} {2k} \\ {k - 1} \\ \end{array}} \right) \left( { \begin{array}{l} {3k} \\ {k - 1} \\ \end{array} } \right) }}{{{{27}^k}}}} \end{aligned}$$

modulo \(p^2\), where \(C_k=\frac{1}{k+1}\left( {\begin{array}{c}2k\\ k\end{array}}\right) \) is the k-th Catalan number and \(C_k^{(2)}=\frac{1}{2k+1}\left( {\begin{array}{c}3k\\ k\end{array}}\right) \) is the second-order Catalan numbers of the first kind. And we prove that

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{D_k}{k}\equiv -q_p(2)+pq_p(2)^2\pmod {p^2}, \end{aligned}$$

where \(D_n=\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}n+k\\ k\end{array}}\right) \) is the n-th Delannoy number and \(q_p(2)=(2^{{p-1}}-1)/p\) is the Fermat quotient.

     

On the Diophantine equation $$X^{2N}+2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5$$

We prove that for each prime p, positive integer \(\alpha \), and non-negative integers \(\beta \) and \(\gamma \), the Diophantine equation \(X^{2N} + 2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5\) has no solution with N, X, \(Z\in \mathbb {Z}^+\), \(N > 1\), and \(\gcd (X,Z) = 1\).     

Congruences for 5-regular partitions modulo powers of 5

Let \(b_{5}(n)\) denote the number of 5-regular partitions of n. We find the generating functions of \(b_{5}(An+B)\) for some special pairs of integers (A, B). Moreover, we obtain infinite families of congruences for \(b_{5}(n)\) modulo powers of 5. For example, for any integers \(k\ge 1\) and \(n\ge 0\), we prove that

$$\begin{aligned} b_{5}\left( 5^{2k-1}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}) \end{aligned}$$

and

$$\begin{aligned} b_{5}\left( 5^{2k}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}). \end{aligned}$$

     

Congruences for 7 and 49-regular partitions modulo powers of 7

Let \(b_{k}(n)\) denote the number of k-regular partitions of n. In this paper, we prove Ramanujan-type congruences modulo powers of 7 for \(b_{7}(n)\) and \(b_{49}(n)\). For example, for all \(j\ge 1\) and \(n\ge 0\), we prove that

$$\begin{aligned} b_{7}\Bigg (7^{2j-1}n+\frac{3\cdot 7^{2j-1}-1}{4}\Bigg )\equiv 0\pmod {7^{j}} \end{aligned}$$

and

$$\begin{aligned} b_{49}\Big (7^{j}n+7^{j}-2\Big )\equiv 0\pmod {7^{j}}. \end{aligned}$$

     

On $$\ell $$-regular bipartitions modulo $$\ell $$

Let \(B_\ell (n)\) denote the number of \(\ell \)-regular bipartitions of n. In this paper, we prove several infinite families of congruences satisfied by \(B_\ell (n)\) for \(\ell \in {\{5,7,13\}}\). For example, we show that for all \(\alpha >0\) and \(n\ge 0\),

$$\begin{aligned} B_5\left( 4^\alpha n+\frac{5\times 4^\alpha -2}{6}\right)\equiv & {} 0 \ (\text {mod}\ 5),\\ B_7\left( 5^{8\alpha }n+\displaystyle \frac{5^{8\alpha }-1}{2}\right)\equiv & {} 3^\alpha B_7(n)\ (\text {mod}\ 7) \end{aligned}$$

and

$$\begin{aligned} B_{13}\left( 5^{12\alpha }n+5^{12\alpha }-1\right) \equiv B_{13}(n)\ (\text {mod}\ 13). \end{aligned}$$

     

On the family of elliptic curves $$\varvec{y^2=x^3-m^2x+p^2}$$

In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of \(E_{m,p} : y^2=x^3-m^2x+p^2\), where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of \(E_{m,p}(\mathbb {Q})\) is trivial for both the cases {\(m\ge 1\), \(m\not \equiv 0\pmod 3\)} and {\(m\ge 1\), \(m \equiv 0 \pmod 3\), with \(gcd(m,p)=1\)}. We also show that given any odd prime p and for any positive integer m with \(m\not \equiv 0\pmod 3\) and \(m\equiv 2\pmod {32}\), the lower bound for the rank of \(E_{m,p}(\mathbb {Q})\) is 2. Finally, we find curves of rank 9 in this family.     

Congruences modulo $$10^3$$ for Euler numbers

For a non-negative integer \(n\), let \(E_n\) be the \(n\) th Euler number. In this note, for any positive integer \(n\), we prove the following congruences:

$$\begin{aligned} {\left\{ \begin{array}{ll} E_{4n} \equiv 380n-375 \pmod {10^3}, \\ E_{4n+2} \equiv -460n+399 \pmod {10^3}. \end{array}\right. } \end{aligned}$$

Our proof is based on induction on \(n\) and elementary direct calculations.

     

Least energy sign-changing solutions for a class of fourth order Kirchhoff-type equations in $$\mathbb {R}^{N}$$

In this article we study the problem

$$\begin{aligned} \Delta ^{2}u-\left( a+b\int _{\mathbb {R}^{N}}\left| \nabla u\right| ^{2}dx\right) \Delta u+V(x)u=\left| u\right| ^{p-2}u\ \text { in }\mathbb {R}^{N}, \end{aligned}$$

where \(\Delta ^{2}:=\Delta (\Delta )\) is the biharmonic operator, \(a,b>0\) are constants, \(N\le 7,\) \(p\in (4,2_{*})\) for \(2_{*}\) defined below, and \(V(x)\in C(\mathbb {R}^{N},\mathbb {R})\). Under appropriate assumptions on V(x), the existence of least energy sign-changing solution is obtained by combining the variational methods and the Nehari method.

     

Global $$C^{1,\al pha }$$ regularity and e xistence of multi ple solutions for singular p( x)-La placian equations

In this paper we study the following singular p(x)-Laplacian problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} - \text{ div } \left( |\nabla u|^{p(x)-2} \nabla u\right) =\frac{ \lambda }{u^{\beta (x)}}+u^{q(x)}, &{} \text{ in }\quad \Omega , \\ u>0, &{} \text{ in }\quad \Omega , \\ u=0, &{} \text{ on }\quad \partial \Omega , \end{array}\right. \end{aligned}$$

where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 2\), with smooth boundary \(\partial \Omega \), \(\beta \in C^1(\bar{\Omega })\) with \( 0< \beta (x) <1\), \(p\in C^1(\bar{\Omega })\), \(q \in C(\bar{\Omega })\) with \(p(x)>1\), \(p(x)< q(x) +1 <p^*(x)\) for \(x \in \bar{\Omega }\), where \( p^*(x)= \frac{Np(x)}{N-p(x)} \) for \(p(x) <N\) and \( p^*(x)= \infty \) for \( p(x) \ge N\). We establish \(C^{1,\alpha }\) regularity of weak solutions of the problem and strong comparison principle. Based on these two results, we prove the existence of multiple (at least two) positive solutions for a certain range of \(\lambda \).

     

Arithmetic properties for a partition function related to the Ramanujan/Watson mock theta function $$\omega (q)$$

Recently, Andrews, Dixit, and Yee introduced a new partition function \(p_{\omega }(n)\) that denotes the number of partitions of n in which each odd part is less than twice the smallest part. The generating function of \(p_{\omega }(n)\) is associated with the third-order mock theta function \(\omega (q)\). Andrews, Passary, Sellers, and Yee proved three infinite families of congruences modulo 4 and 8 for \(p_{\omega }(n)\) and provided elementary proofs of congruences modulo 5 for \(p_{\omega }(n)\) which were first discovered by Waldherr. In this paper, we prove some new congruences modulo 5 and powers of 2 for \(p_{\omega }(n)\). In particular, we obtain some non-standard congruences for \(p_{\omega }(n)\). For example, we prove that for \(k\ge 0\), \( p_{\omega }\left( \frac{7\times 5^{2k+1}+1 }{3}\right) \equiv (-1)^k \ (\mathrm{mod}\ 5) \) and \( p_\omega \left( \frac{2^{2k+7}+1}{3}\right) \equiv 1251 \times (-1)^k \ (\mathrm{mod}\ 2^{11})\).     

Congruences for (3,11)-regular bipartitions modulo 11

In this note we investigate the function \(B_{k,\ell }(n)\), which counts the number of \((k,\ell )\)-regular bipartitions of n. We shall prove an infinite family of congruences modulo 11: for \(\alpha \ge 2\) and \(n\ge 0\),

$$\begin{aligned} B_{3,11}\left( 3^{\alpha }n+\frac{5\cdot 3^{\alpha -1}-1}{2}\right) \equiv 0\ (\mathrm{mod\ }11). \end{aligned}$$

     

A variation of a congruence of Subbarao for $$n=2^{\alpha }5^{\beta }$$

There are many open problems concerning the characterization of the positive integers n fulfilling certain congruences and involving the Euler totient function \(\varphi \) and the sum of positive divisors function \(\sigma \) of the positive integer n. In this work, we deal with the congruence of the form

$$\begin{aligned} n\varphi (n)\equiv 2\quad \pmod {\sigma (n)}, \end{aligned}$$

and prove that the only positive integers of the form \(2^{\alpha }5^{\beta },\, \alpha , \,\beta \ge 0,\) that satisfy the above congruence are \(n=1,\, 2,\, 5,\, 8.\)

     

Arithmetic properties of 7-regular partitions

Let \(b_{\ell }(n)\) denote the number of \(\ell \)-regular partitions of n. By employing the modular equation of seventh order, we establish the following congruence for \(b_{7}(n)\) modulo powers of 7: for \(n\ge 0\) and \(j\ge 1\),

$$\begin{aligned} b_{7}\left( 7^{2j-1}n+\frac{3\cdot 7^{2j}-1}{4}\right) \equiv 0 \pmod {7^j}. \end{aligned}$$

We also find some infinite families of congruences modulo 2 and 7 satisfied by \(b_{7}(n)\).

     

On solutions of the simultaneous Pell equations $$ x^{2}-\left( a^{2}-1\right) y^{2}=1$$ and $$y^{2}-pz^{2}=1$$y2-pz2=1

Let \(a\ge 2\) be an integer and p prime number. It is well-known that the solutions of the Pell equation have recurrence relations. For the simultaneous Pell equations

$$\begin{aligned}&x^{2}-\left( a^{2}-1\right) y^{2} =1 \\&y^{2}-pz^{2} =1 \end{aligned}$$

assume that \(x=x_{m}\) and \(y=y_{m}\). In this paper, we show that if \(m\ge 3\) is an odd integer, then there is no positive solution to the system. Moreover, we find the solutions completely for \(5\le a\le 14\) in the cases when \(m\ge 2\) is even integer and \(m=1\).

     

Local uniqueness and periodicity for the prescribed scalar curvature problem of fractional operator in $${\mathbb {R}}^{N}$$

Consider the following prescribed scalar curvature problem involving the fractional Laplacian with critical exponent:

$$\begin{aligned} \left\{ \begin{array}{ll}(-\Delta )^{\sigma }u=K(y)u^{\frac{N+2\sigma }{N-2\sigma }} \text { in }~ {\mathbb {R}}^{N},\\ ~u>0, \quad y\in {\mathbb {R}}^{N}.\end{array}\right. \end{aligned}$$

(0.1)

For \(N\ge 4\) and \(\sigma \in (\frac{1}{2}, 1),\) we prove a local uniqueness result for bubbling solutions of (0.1). Such a result implies that some bubbling solutions preserve the symmetry from the scalar curvature K(y).

     

New constructions of permutation polynomials of the form $$x^rh\left( x^{q-1}\right) $$ over $${\mathbb F}_{q^2}$$Fq2

Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation trinomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\), where \(q=2^k\), \(h(x)=1+x^s+x^t\) and \(r, k>0, s, t\) are integers. Their methods are essentially usage of a multiplicative version of AGW Criterion because they all transformed the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing the corresponding fractional polynomials permute a smaller set \(\mu _{q+1}\), where \(\mu _{q+1}:=\{x\in \mathbb {F}_{q^2} : x^{q+1}=1\}\). Motivated by these results, we characterize the permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\) such that \(h(x)\in {\mathbb F}_q[x]\) is arbitrary and q is also an arbitrary prime power. Using AGW Criterion twice, one is multiplicative and the other is additive, we reduce the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing permutations over a small subset S of a proper subfield \({\mathbb F}_{q}\), which is significantly different from previously known methods. In particular, we demonstrate our method by constructing many new explicit classes of permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\). Moreover, we can explain most of the known permutation trinomials, which are in Ding et al. (SIAM J Discret Math 29:79–92, 2015), Gupta and Sharma (Finite Fields Appl 41:89–96, 2016), Li and Helleseth (Cryptogr Commun 9:693–705, 2017), Li et al. (New permutation trinomials constructed from fractional polynomials, arXiv: 1605.06216v1, 2016), Li et al. (Finite Fields Appl 43:69–85, 2017) and Zha et al. (Finite Fields Appl 45:43–52, 2017) over finite field with even characteristic.     

Tauberian conditions for the $$(C, \alpha )$$ integrability of functions

For a real-valued continuous function f(x) on \([0,\infty )\), we define

$$\begin{aligned} s(x)=\int _{0}^{x} f(u)du\quad \text {and}\quad \sigma _{\alpha } (x)= \int _{0}^{x}\left( 1-\frac{u}{x}\right) ^{\alpha }f(u)du \end{aligned}$$

for \(x>0\). We say that \(\int _{0}^{\infty } f(u)du\) is \((C, \alpha )\) integrable to L for some \(\alpha >-1\) if the limit \(\lim _{x \rightarrow \infty } \sigma _{\alpha } (x)=L\) exists. It is known that \(\lim _{x \rightarrow \infty } s(x) =L\) implies \(\lim _{x \rightarrow \infty }\sigma _{\alpha } (x) =L\) for all \(\alpha >-1\). The aim of this paper is twofold. First, we introduce some new Tauberian conditions for the \((C, \alpha )\) integrability method under which the converse implication is satisfied, and improve classical Tauberian theorems for the \((C,\alpha )\) integrability method. Next we give short proofs of some classical Tauberian theorems as special cases of some of our results.

     

Blow Up Criteria for the Incompressible Nematic Liquid Crystal Flows

In this paper, we investigate blow up criteria for the local smooth solutions to the 3D incompressible nematic liquid crystal flows via the components of the gradient velocity field \(\nabla u\) and the gradient orientation field \(\nabla d\). More precisely, we show that \(0< T_{ \ast}<+\infty\) is the maximal time interval if and only if

$$\begin{aligned} & \int_{0}^{T_{\ast}} \bigl\Vert \Vert \partial_{i}u\Vert _{L_{x_{i}} ^{\gamma}} \bigr\Vert _{L_{x_{j}x_{k}}^{\alpha}}^{\beta}+ \|\nabla d\| _{L^{\infty}}^{\frac{8}{3}}\mathrm{d}t=\infty, \\ &\quad\text{ with } \frac{2}{\alpha}+\frac{2}{\beta}\leq\frac{3\alpha +2}{4\alpha}, \text{ and } 1\leq\gamma\leq\alpha,2< \alpha\leq+\infty, \end{aligned}$$

or

$$\begin{aligned} \int_{0}^{T_{\ast}}\|\partial_{3}u_{3} \|^{\beta}_{L^{\alpha}}+\| \nabla d\|^{\frac{8}{3}}_{L^{\infty}} \mathrm{d}t=\infty,\quad\text{with } \frac{3}{\alpha}+\frac{2}{\beta}\leq \frac{3(\alpha+2)}{4 \alpha}, \text{ and } 2< \alpha\leq\infty, \end{aligned}$$

where \(i,j,k\in\{1,2,3\}\), \(i\neq j\), \(i\neq k\), and \(j\neq k\).

     

On some constacyclic codes over $$\mathbb {Z}_{4}\left[ u\right] /\left\langle u^{2}-1\right\rangle $$, their $$\mathbb {Z}_4$$Z4 images,and new codes

In this paper, we study \(\lambda \)-constacyclic codes over the ring \(R=\mathbb {Z}_4+u\mathbb {Z}_4\) where \(u^{2}=1\), for \(\lambda =3+2u\) and \(2+3u\). Two new Gray maps from R to \(\mathbb {Z}_4^{3}\) are defined with the goal of obtaining new linear codes over \(\mathbb {Z}_4\). The Gray images of \(\lambda \)-constacyclic codes over R are determined. We then conducted a computer search and obtained many \(\lambda \)-constacyclic codes over R whose \(\mathbb {Z}_4\)-images have better parameters than currently best-known linear codes over \(\mathbb {Z}_4\).