相关论文: On a conjecture of Erd\"os and certain Dirichlet s…

In a written correspondence with A. Livingston, Erd\H{o}s conjectured that for any arithmetical function $f$, periodic with period $q$, taking values in $\{-1,1\}$ when $q \nmid n$ and $f(n)=0$ when $q \mid n$, the series…

Erd\H{o}s proved that $\mathcal{F}(A) := \sum_{a \in A}\frac{1}{a\log a}$ converges for any primitive set of integers $A$ and later conjectured this sum is maximized when $A$ is the set of primes. Banks and Martin further conjectured that…

Let $\sigma(x)$ be the sum of the divisors of $x$. If $N$ is odd and $\sigma(N) = 2N$, then the odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and…

Let $q\geq 2$ and denote by $s_q$ the sum-of-digits function in base $q$. For $j=0,1,...,q-1$ consider $$# \{0 \le n < N : \;\;s_q(2n) \equiv j \pmod q \}.$$ In 1983, F. M. Dekking conjectured that this quantity is greater than $N/q$ and,…

In number theory, the Erdos-Straus conjecture states that for all n >=2, the rational number 4/n can be expressed as the sum of three unit fractions. Paul Erdos and Ernst G. Straus formulated the conjecture in 1948. The restriction that the…

The Erd\H{o}s-Straus conjecture is a renowned problem which describes that for every natural number $n~(\ge 2)$, $\frac{4}{n}$ can be represented as the sum of three unit fractions. The main purpose of this study is to show that the…

Euler showed that if an odd perfect number $N$ exists, it must consist of two parts $N=q^k n^2$, with $q$ prime, $q \equiv k \equiv 1 \pmod{4}$, and gcd$(q,n)=1$. Dris conjectured that $q^k < n$. We first show that $q<n$ for all odd perfect…

In this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer $n>1$ and $M=(n+1)/2$ or $n-1$, $$ \sum_{k=0}^{M}[4k-1]_{q^2}[4k-1]^2\frac{(q^{-2};q^4)_k^4}{(q^4;q^4)_k^4}q^{4k}\equiv…

Dris conjectured in his masters thesis that the inequality $q^k < n$ always holds, if $N = {q^k}{n^2}$ is an odd perfect number with special prime $q$. In this note, we initially show that either of the two conditions $n < q^k$ or…

A strictly increasing sequence $\mathscr{A}$ of positive integers is said to be primitive if no term of $\mathscr{A}$ divides any other. Erd\H{o}s showed that the series $\sum_{a \in \mathscr{A}} \frac{1}{a \log a}$, where $\mathscr{A}$ is…

In 1992 Strauss, Shallit and Zagier proved that for any positive integer $a$ we have $$\sum_{k=0}^{3^a-1}\binom{2k}{k}=0 (mod 3^{2a})$$ and furthermore $$3^{-2a}}\sum_{k=0}^{3^a-1}\binom{2k}k=1 (mod 3).$$ Recently a $q$-analogue of the…

A famous conjecture of Erd\H os and Straus is that for every integer $n\ge2$, $4/n$ can be represented as $1/x+1/y+1/z$, where $x,y,z$ are positive integers. This conjecture was generalized to $5/n$ by Sierpi\'nski, and then Schinzel…

The Erd\"{o}s--Straus conjecture states that the equation $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ has positive integer solutions $x,y,z$ for every postive integers $n\geq 2$. In this short note we find explicity the solutions of…

Let \[ \sum_{n=0}^{\infty}A(n)q^{n} := \frac{(q^{2};q^{5})_{\infty}^{5}(q^{3};q^{5})_{\infty}^{5}}{(q;q^{5})_{\infty}^{5}(q^{4};q^{5})_{\infty}^{5}}, \] \[ \sum_{n=0}^{\infty} B(n)q^{n} := \frac{(q;q^{5})_{\infty}^{5}…

From the Rhind Papyrus and other extant sources, we know that the ancient Egyptians were very iterested in expressing a given fraction into a sum of unit fractions, that is fractions whose numerators are equal to 1. One of the problems that…

For any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ with $x,y,z$ positive integers. The \emph{Erd\H{o}s-Straus conjecture} asserts that…

Let $a(n)$ defined by $\sum_{n=1}^{\infty}a(n)q^n := \prod_{n=1}^{\infty}\frac{1}{(1-q^{3n})(1-q^n)^3}.$ In this note, we prove that for every non-negative integer $n$, a(15n+6) \equiv 0\pmod{5}, a(15n+12) \equiv 0\pmod{5}. As a corollary,…

Let $f(n)$ be a multiplicative function satisfying $|f(n)|\leq 1$, $q$ $(\leq N^2)$ be a prime number and $a$ be an integer with $(a,\,q)=1$, $\chi$ be a non-principal Dirichlet character modulo $q$. In this paper, we shall prove that $$…

Let $N$ be an odd perfect number. Then, Euler proved that there exist some integers $n, \alpha$ and a prime $q$ such that $N = n^{2}q^{\alpha}$, $q \nmid n$, and $q \equiv \alpha \equiv 1 \bmod 4$. In this note, we prove that the ratio…

Let $f(n)$ count the number of subsets of $\{1,...,n\}$ without an element dividing another. In this paper I show that $f(n)$ grows like the $n$-th power of some real number, in the sense that $\lim_{n\rightarrow \infty}f(n)^{1/n}$ exists.…