A variety of Euler's conjecture

We consider a variety of Euler's conjecture, i.e., whether the Diophantine system $$\begin{cases} n=a_{1}+a_{2}+\cdots+a_{s-1}, a_{1}a_{2}\cdots a_{s-1}(a_{1}+a_{2}+\cdots+a_{s-1})=b^{s} \end{cases}$$ has solutions $n,b,a_i\in\mathbb{Z}^+,i=1,2,\ldots,s-1,s\geq 3.$ By using the theory of elliptic curves, we prove that it has no solutions $n,b,a_i\in\mathbb{Z}^+$ for $s=3$, but for $s=4$ it has infinitely many solutions $n,b,a_i\in\mathbb{Z}^+$ and for $s\geq 5$ there are infinitely many polynomial solutions $n,b,a_i\in\mathbb{Z}[t_1,t_2,\ldots,t_{s-3}]$ with positive value satisfying this Diophantine system.Comment: 8 page

Super congruences involving alternating harmonic sums modulo prime powers

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 establish a combinational congruence of alternating harmonic sums for any odd prime $p$ and positive integers $r$, \begin{equation*} \sum\limits_{i+j+k=p^{r}\atop{i,j,k\in \mathcal{P}_{p}}}\frac{(-1)^{i}}{ijk} \equiv \frac{1}{2}p^{r-1}B_{p-3} (\bmod p^{r}). \end{equation*} For any odd prime $p\geq 5$ and positive integers $r$, we have \begin{align} &4\sum\limits_{i_{1}+i_{2}+i_{3}+i_{4}=2p^{r}\atop{i_{1}, i_{2}, i_{3}, i_{4}\in \mathcal{P}_{p}}}\frac{(-1)^{i_{1}}}{i_{1}i_{2}i_{3}i_{4}}+3\sum\limits_{i_{1}+i_{2}+i_{3}+i_{4}=2p^{r}\atop{i_{1}, i_{2}, i_{3}, i_{4}\in \mathcal{P}_{p}}}\frac{(-1)^{i_{1}+i_{2}}}{i_{1}i_{2}i_{3}i_{4}} \nonumber\\&\equiv\begin{cases} \frac{216}{5}pB_{p-5}\pmod{p^{2}}, if r=1, \\ \frac{36}{5}p^{r}B_{p-5}\pmod{p^{r+1}}, if r>1. \\ \end{cases}\nonumber \end{align} For any odd prime $p> 5$ and positive integers $r$, we have \begin{align} &\sum\limits_{i_{1}+i_{2}+i_{3}+i_{4}+i_{5}=2p^{r}\atop{i_{1}, i_{2}, i_{3}, i_{4}, i_{5}\in \mathcal{P}_{p}}}\frac{(-1)^{i_{1}}}{i_{1}i_{2}i_{3}i_{4}i_{5}}+2\sum\limits_{i_{1}+i_{2}+i_{3}+i_{4}+i_{5}=2p^{r}\atop{i_{1}, i_{2}, i_{3}, i_{4}, i_{5}\in \mathcal{P}_{p}}}\frac{(-1)^{i_{1}+i_{2}}}{i_{1}i_{2}i_{3}i_{4}i_{5}} \nonumber\\&\equiv\begin{cases} 12B_{p-5}\pmod{p}, if r=1,\\ 6p^{r-1}B_{p-5}\pmod{p^{r}}, if r>1. \end{cases}\nonumber \end{align

Perfect numbers and Fibonacci primes (III)

In this article, we consider the Diophantine equation $\sigma_{2}(n)-n^2=An+B$ with $A=P^2\pm2$. For some $B$, we show that except for finitely many computable solutions in the range $n\leq(|A|+|B|)^{3}$, all the solutions are expressible in terms of Lucas sequences. Meanwhile, we obtain some results relating to other linear recurrent sequences.Comment: 11 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

On the number of representations of n=a+bn=a+b with abab a multiple of a polygonal number

In this paper, we study the number of representations of a positive integer $n$ by two positive integers whose product is a multiple of a polygonal number.Comment: 8 page

On the Lucas Property of Linear Recurrent Sequences

We say that an arithmetical function $S:\mathbb{N}\rightarrow\mathbb{Z}$ has Lucas property if for any prime $p$, \begin{equation*} S(n)\equiv S(n_{0})S(n_{1})\ldots S(n_{r})\pmod p, \end{equation*} where $n=\sum_{i=0}^{r}n_{i}p^{i}$, with $0 \leq n_{i} \leq p-1,n,n_{i}\in\mathbb{N}$. In this note, we discuss the Lucas property of Fibonacci sequences and Lucas numbers. Meanwhile, we find some other interesting results.Comment: 8 page

Congruent numbers on the right trapezoid

We introduce and study a new kind of congruent number problem on the right trapezoid.Comment: 12 pages, 2 figure

A congruence involving 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*} Z(p^{r})\equiv-2p^{r-1}B_{p-3} ~(\bmod ~ p^{r}), \end{equation*} where $Z(n)=\sum\limits_{i+j+k=n\atop{i,j,k\in\mathcal{P}_{n}}}\frac{1}{ijk}$ and $\mathcal{P}_{n}$ denote the set of positive integers which are prime to $n$. In this note, we obtain a congruence for distinct odd primes $p,~q$ and positive integers $\alpha,~\beta$, \begin{equation*} Z(p^{\alpha}q^{\beta})\equiv 2(2-q)(1-\frac{1}{q^{3}})p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{\alpha}} \end{equation*} and the necessary and sufficient condition for \begin{equation*} Z(p^{\alpha}q^{\beta})\equiv 0\pmod{p^{\alpha}q^{\beta}}. \end{equation*} Finally, we raise a conjecture that for $n>1$ and odd prime power $p^{\alpha}||n$, $\alpha\geq1$, \begin{eqnarray} \nonumber Z(n)\equiv \prod\limits_{q|n\atop{q\neq p}}(1-\frac{2}{q})(1-\frac{1}{q^{3}})(-\frac{2n}{p})B_{p-3}\pmod{p^{\alpha}}. \end{eqnarray

A New Generalization of Fermat's Last Theorem

In this paper, we consider some hybrid Diophantine equations of addition and multiplication. We first improve a result on new Hilbert-Waring problem. Then we consider the equation \begin{equation} \begin{cases} A+B=C ABC=D^n \end{cases} \end{equation} where A,B,C,D,n \in\ZZ_{+} and $n\geq3$, which may be regarded as a generalization of Fermat's equation $x^n+y^n=z^n$. When $\gcd(A,B,C)=1$, $(1)$ is equivalent to Fermat's equation, which means it has no positive integer solutions. We discuss several cases for $\gcd(A,B,C)=p^k$ where $p$ is an odd prime. In particular, for $k=1$ we prove that $(1)$ has no nonzero integer solutions when $n=3$ and we conjecture that it is also true for any prime $n>3$. Finally, we consider equation $(1)$ in quadratic fields $\mathbb{Q}(\sqrt{t})$ for $n=3$.Comment: 11 pages, revise

A congruence involving the quotients of Euler and its applications (III)

In this paper, we will present several new congruences involving binomial coefficients under integer moduli, which are the continuation of the previous two work by Cai \textit{et al.} (2002, 2007).Comment: 13 page