论文标题
关于与Sierpiński和Riesel编号相关的二项式系数
On binomial coefficients associated with Sierpiński and Riesel numbers
论文作者
论文摘要
在本文中，我们研究了Sierpiński数字和Riesel数字作为二项式系数的存在。我们表明，对于任何奇怪的积极整数$ r $，都存在$ \ binom {k} {r} $的许多Sierpiński编号和Riesel编号。令$ s（x）$为正整数的数量$ r $满足$ 1 \ leq r \ leq x $，$ \ binom {k} {r} {r} $是无限的$ k $的sierpiński数字。我们进一步表明，$ s（x）/x $的值随着$ x $倾向于无穷大。还考虑了对$ a $-sierpiński数字和$ a $ a-riesel号码的概括。特别是，我们证明存在无限的许多正整数$ r $，因此$ \ binom {k} {r} $同时是一个基础$ a $ a $-sierpiński和base $ a $ rieseel number，用于无限的$ k $。
In this paper, we investigate the existence of Sierpiński numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer $r$, there exist infinitely many Sierpiński numbers and Riesel numbers of the form $\binom{k}{r}$. Let $S(x)$ be the number of positive integers $r$ satisfying $1\leq r\leq x$ for which $\binom{k}{r}$ is a Sierpiński number for infinitely many $k$. We further show that the value $S(x)/x$ gets arbitrarily close to 1 as $x$ tends to infinity. Generalizations to base $a$-Sierpiński numbers and base $a$-Riesel numbers are also considered. In particular, we prove that there exist infinitely many positive integers $r$ such that $\binom{k}{r}$ is simultaneously a base $a$-Sierpiński and base $a$-Riesel number for infinitely many $k$.
