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In this paper, we investigate the existence of Sierpi{n}ski numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer $r$, there exist infinitely many Sierpi{n}ski numbers and Riesel numbers of the form $binom{k}{r}$. Let $S(x)$ be the number of positive integers $r$ satisfying $1leq rleq x$ for which $binom{k}{r}$ is a Sierpi{n}ski 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 $a$-Sierpi{n}ski 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 $a$-Sierpi{n}ski and base $a$-Riesel number for infinitely many $k$.
Phase transition of the classical Ising model on the Sierpi{n}ski carpet, which has the fractal dimension $log_3^{~} 8 approx 1.8927$, is studied by an adapted variant of the higher-order tensor renormalization group method. The second-order phase tr
Let $p$ be a prime with $p>3$, and let $a,b$ be two rational $p-$integers. In this paper we present general congruences for $sum_{k=0}^{p-1}binom akbinom{-1-a}kfrac p{k+b}pmod {p^2}$. For $n=0,1,2,ldots$ let $D_n$ and $b_n$ be Domb and Almkvist-Zudil
We present several sequences involving harmonic numbers and the central binomial coefficients. The calculational technique is consists of a special summation method that allows, based on proper two-valued integer functions, to calculate different fam
The feedback vertex number $tau(G)$ of a graph $G$ is the minimum number of vertices that can be deleted from $G$ such that the resultant graph does not contain a cycle. We show that $tau(S_p^n)=p^{n-1}(p-2)$ for the Sierpi{n}ski graph $S_p^n$ with $
We propose higher-order generalizations of Jacobsthals $p$-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients $binom{ip}{p}$ ($i=1,2,dots$) that are divisible by arbitrarily large powers of prime $p$.