In 1909, Hilbert proved that for each fixed k, there is a number g with the following property: Every integer N ≥ 0 has a representation in the form N = x 1k + x 2k + … + x gk, where the x i are nonnegative integers. This resolved a conjecture of Edward Waring from 1770. Hilbert’s proof is somewhat unsatisfying, in that no method is given for finding a value of g corresponding to a given k. In his doctoral thesis, Rieger showed that by a suitable modification of Hilbert’s proof, one can give explicit...

A set $𝒮=\{x_1,...,x_n\}$ of $n$ distinct positive integers is said to be gcd-closed if $(x_i,x_j)\in 𝒮$ for all $1\le i,j\le n$. Shaofang Hong conjectured in 2002 that for a given positive integer $t$ there is a positive integer $k\left(t\right)$ depending only on $t$, such that if $n\le k\left(t\right)$, then the power LCM matrix $\left({[x_i,x_j]}^t\right)$ defined on any gcd-closed set $𝒮=\{x_1,...,x_n\}$ is nonsingular, but for $n\ge k\left(t\right)+1$, there exists a gcd-closed set $𝒮=\{x_1,...,x_n\}$ such that the power LCM matrix $\left({[x_i,x_j]}^t\right)$ on $𝒮$ is singular. In 1996, Hong proved $k\left(1\right)=7$ and noted $k\left(t\right)\ge 7$ for all $t\ge 2$. This paper develops Hong’s method and provides a new idea to calculate...

In 1989, E. Saias established an asymptotic formula for $\Psi (x,y)=\left|\left\{n\le x:p\mid n\Rightarrow p\le y\right\}\right|$ with a very good error term, valid for $exp\left({(loglogx)}^{(5/3)+ϵ}\right)\le y\le x$, $x\ge x_0\left(ϵ\right)$, $ϵ\>0.$ We extend this result to an algebraic number field $K$ by obtaining an asymptotic formula for the analogous function ${\Psi }_K(x,y)$ with the same error term and valid in the same region. Our main objective is to compare the formulae for $\Psi (x,y)$ and ${\Psi }_K(x,y),$ and in particular to compare the second term in the two expansions.

We obtain the values concerning $(\theta ,\varphi )=limin{f}_{\left|q\right|\to \infty }\left|q\right|\Vert q\theta -\varphi \Vert$ using the algorithm by Nishioka, Shiokawa and Tamura. In application, we give the values (θ,1/2), (θ,1/a), (θ,1/√(ab(ab+4))) and so on when θ = (√(ab(ab+4)) - ab)/(2a) = [0;a,b,a,b,...].

We consider the values concerning$$ℳ(\theta ,\phi )=\underset{\left|q\right|\to \infty }{lim\; inf}\left|q\right|\left|\right|q^{\theta }-\phi \left|\right|$$where the continued fraction expansion of $\theta$ has a quasi-periodic form. In particular, we treat the cases so that each quasi-periodic form includes no constant. Furthermore, we give some general conditions satisfying $ℳ(\theta ,\phi )=0$.