论文标题
关于格雷厄姆在中央二项式系数的p-划分性上的猜想
On a conjecture of Graham on the p-divisibility of central binomial coefficients
论文作者
论文摘要
我们表明，对于每$ r \ geq 1 $，以及所有$ r $ DISTINT（足够大）$ p_1，...，p_r> p_0（r）$，存在无限的许多整数$ n $，因此$ {2n \ select n} $仅由这些刺激性排除，而这些普rime to plimes to pollime forly Down tollow solly low flowly solowy lowlliplicity均可将其排除。从kummer定理中，上限的上限，用于prime $ p_j $可以划分$ {2n \ select n} $的次数为$ 1+\ log n / \ log n / \ log log p_j $;并且我们的定理表明，对于每$ \ varepsilon> 0 $，$ r \ geq 1 $，以及任何足够大的素数$ p_1，...，p_r> p_0（\ varepsilon，r）$，我们可以找到整数$ n $ n $ n $ n $ n $ j = 1，$ j = 1，... $ \ varepsilon \ log n/\ log p_j $。我们将此结果与R. L. Graham的著名猜想联系起来，即是否有无限的整数$ n $，以至于$ {2n \ select n} $ coprime是$ 105 $。
We show that for every $r \geq 1$, and all $r$ distinct (sufficiently large) primes $p_1,..., p_r > p_0(r)$, there exist infinitely many integers $n$ such that ${2n \choose n}$ is divisible by these primes to only low multiplicity. From a theorem of Kummer, an upper bound for the number of times that a prime $p_j$ can divide ${2n \choose n}$ is $1+\log n / \log p_j$; and our theorem shows that for every $\varepsilon > 0$, $r \geq 1$, and any sufficiently large primes $p_1,...,p_r > p_0(\varepsilon,r)$, we can find integers $n$ where for $j=1,...,r$, $p_j$ divides ${2n \choose n}$ with multiplicity at most $\varepsilon \log n/\log p_j$. We connect this result to a famous conjecture by R. L. Graham on whether there are infinitely many integers $n$ such that ${2n \choose n}$ is coprime to $105$.
