关于与Sierpiński数和Riesel数相关的二项式系数
On binomial coefficients associated with Sierpiński and Riesel numbers
AI总结：
本文证明任意奇正整数r对应无穷多个形如二项式系数的Sierpiński数和Riesel数，并推广到基a情形，同时给出计数函数的渐近性质。
摘要：
本文研究了二项式系数中Sierpiński数和Riesel数的存在性。我们证明，对于任意奇正整数$r$，存在无穷多个形如$\binom{k}{r}$的Sierpiński数和Riesel数。设$S(x)$为满足$1\leq r\leq x$且$\binom{k}{r}$对无穷多个$k$为Sierpiński数的正整数$r$的个数。我们进一步证明，当$x$趋于无穷时，$S(x)/x$的值任意接近1。我们还考虑了基$a$-Sierpiński数和基$a$-Riesel数的推广。特别地，我们证明存在无穷多个正整数$r$，使得$\binom{k}{r}$同时是基$a$-Sierpiński数和基$a$-Riesel数，且对无穷多个$k$成立。
英文摘要：
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 $k$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
