arXiv subjects

Yong-Gao Chen

Publications and source records attributed to Yong-Gao Chen.

25 records · Page 2Linked to original sources

arXiv2012

Weighted representation functions on $\mathbb{Z}_m$

Let $m$, $k_1$, and $k_2$ be three integers with $m\ge 2$. For any set $A\subseteq \mathbb{Z}_m$ and $n\in \mathbb{Z}_m$, let $\hat{r}_{k_1,k_2}(A,n)$ denote the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. In this paper, using exponential sums, we characterize all $m$, $k_1$, $k_2$, and $A$ for which $\hat{r}_{k_1,k_2}(A,n)=\hat{r}_{k_1,k_2}(\mathbb{Z}_m\setminus A,n)$ for all $n\in \mathbb{Z}_m$. We also pose several problems for further research.

math.NT↗

arXiv2012

On a problem of Sierpinski

Let $s\ge 2$ be an integer. Denote by $μ_s$ the least integer so that every integer $\ell >μ_s$ is the sum of exactly $s$ integers $>1 $ which are pairwise relatively prime. In 1964, Sierpiński asked a determination of $μ_s$. Let $p_1=2$, $p_2=3, ...$ be the sequence of consecutive primes and let $μ_s = p_2+p_3+...+p_{s+1}+c_s$. P. Erd\H os proved that there exists an absolute constant $C$ with $-2\le c_s\le C$. In this paper, we determine $μ_s$ for all $s\ge 2$. As a corollary, we show that $-2\le c_s\le 1100$ and the set of integers $s$ with $μ_s= p_2+p_3+... +p_{s+1}+1100$ has the asymptotic density 1.

math.NT↗

arXiv2011

On a Problem of Erdős, Herzog and Schönheim

Let $p_1, p_2,..., p_n$ be distinct primes. In 1970, Erd\H os, Herzog and Schönheim proved that if $\cal D$ is a set of divisors of $N=p_1^{α_1}...p_n^{α_n}$, $α_1\ge α_2\ge...\ge α_n$, no two members of the set being coprime and if no additional member may be included in $\cal D$ without contradicting this requirement then $ |{\cal D}|\ge α_n \prod_{i=1}^{n-1} (α_i +1)$. They asked to determine all sets $\cal D$ such that the equality holds. In this paper we solve this problem. We also pose several open problems for further research.

math.CO↗

arXiv2011

On Sequences Containing at Most 4 Pairwise Coprime Integers

Let $f(n,k)$ be the largest number of positive integers not exceeding $n$ from which one cannot select $k+1$ pairwise coprime integers, and let $E(n,k)$ be the set of positive integers which do not exceed $n$ and can be divided by at least one of $p_1, p_2,..., p_k$, where $p_i$ is the $i$-th prime. In 1962, P. Erd\H os conjectured that $f(n,k)=|E(n,k)|$ for all $n\ge p_k$. In 1973, S. L. G. Choi proved that the conjecture is true for $k=3$. In 1994, Ahlswede and Kachatrian disproved the conjecture for $k=212$. In this paper we prove that, for $n\ge 49$, if A(n,4) is a set of positive integers not exceeding $n$ from which one cannot select 5 pairwise coprime integers and $|A(n,4)|\ge |E(n,4)|$, then $A(n,4)=E(n,4)$. In particular, the conjecture is true for k=4. Several open problems and conjectures are posed for further research.

math.NT↗

arXiv2011

On the elementary symmetric functions of $1, 1/2, \ldots , 1/n$

In 1946, P. Erd\H os and I. Niven proved that there are only finitely many positive integers $n$ for which one or more elementary symmetric functions of $1, 1/2, \ldots , 1/n$ are integers. In this paper we solve this old problem by showing that if $n\ge 4$, then none of elementary symmetric functions of $1, 1/2, \ldots , 1/n$ is an integer.

math.NT↗

arXiv2011

Egyptian Fractions with Restrictions

Let $T_o(k)$ denote the number of solutions of $\sum_{i=1}^k\frac 1{x_i}=1$ in odd numbers $1 1$ in $S(p_1, p_2,..., p_t)$ is more than 1. In this paper we study $T_o(k)$ and $T_k(p_1,..., p_t)$. Three of our results are: 1) $T_o(2k+1)\ge (\sqrt 2)^{(k+1)(k-4)}$ for all $k\ge 4$; 2) if the inverse sum of all elements $s_j>1$ in $S(p_1, p_2,..., p_t)$ is more than 1, then $T_k(p_1,..., p_t)\not= 0$ for infinitely many $k$ and the set of these $k$ is the union of finitely many arithmetic progressions; 3) there exists two constants $k_0=k_0(p_1,..., p_t)>1$ and $c=c(p_1,..., p_t)>1$ such that for any $k>k_0$ we have either $T_k(p_1,..., p_t)= 0$ or $T_k(p_1,..., p_t)>c^k$.

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arXiv2005

Prime values of reducible polynomials, II

The Schinzel hypothesis claims (but it seems hopeless to prove) that any irreducible Q[x] polynomial without a constant factor assumes infinitely many prime values at integer places. On the other hand, it is easy to see that a reducible Q[x] polynomial can have only finitely many such places. In this paper we prove that a reducible Z[x] polynomial of degree n (where n is not 4 or 5) can have at most n+2 such places, and there exist examples with n+1 places. If the Schinzel hypothesis and the k-prime-tuple conjecture are true, then there are also polynomials with n+2 such places. (For n=4 or 5 the maximum possible value is 8.) If a Z[x] polynomial is the product of two non-constant integer-valued Q[x] polynomials, then there are at most 1.87234...n+o(n) such places. Even in this case, the number of integer places with positive prime values is at most n. More generally, if f=gh \in R[x] is a product of two non-constant real polynomials, then the number of real places x such that |g(x)|=1 or |h(x)=1|, while f(x)>1, is at most n. For a natural complex version of this last statement we give a counterexample. Furthermore, we briefly consider the generalized problem of assuming values in a fixed finite set of integers.

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