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arXiv 1706.09804math.NT
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伯努利多项式的分母

Denominators of Bernoulli polynomials

Olivier Bordellès, Florian Luca, Pieter Moree, Igor E. Shparlinski

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25 pages
发表信息：
Mathematika 64 (2018) 519-541
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摘要：

对于正整数$n$，设$\mathfrak{P}_n=\prod_{s_p(n)\ge p} p,$，其中$p$遍历所有素数，$s_p(n)$是$n$的基本$p$数字之和。对于所有的$n$，我们证明了$\mathfrak{P}_n$是整除的所有“小”素数，最多有一个例外。我们还证明了$\mathfrak{P}_n$是大的，有许多素因子超过$\sqrt{n}$，最大的一个超过$n^{20/37}$。我们建立了Kellner猜想，它说，超过$\sqrt{n}$的素因子的数量渐近增长为$κ\sqrt{n}/\log n$对于一些常数$κ$与$κ=2$。此外，我们比较了$\mathfrak{P}_n$和$\mathfrak{P}_{n+1}$的大小，得出了一个有点令人惊讶的结论，即虽然$\mathfrak{P}_n$与$n$趋于无穷大，但不等式$\mathfrak{P}_n>\mathfrak{P}_{n+1}$比它的逆更频繁。

英文摘要：

For a positive integer $n$ let $\mathfrak{P}_n=\prod_{s_p(n)\ge p} p,$ where $p$ runs over all primes and $s_p(n)$ is the sum of the base $p$ digits of $n$. For all $n$ we prove that $\mathfrak{P}_n$ is divisible by all "small" primes with at most one exception. We also show that $\mathfrak{P}_n$ is large, has many prime factors exceeding $\sqrt{n}$, with the largest one exceeding $n^{20/37}$. We establish Kellner's conjecture, which says that the number of prime factors exceeding $\sqrt{n}$ grows asymptotically as $κ\sqrt{n}/\log n$ for some constant $κ$ with $κ=2$. Further, we compare the sizes of $\mathfrak{P}_n$ and $\mathfrak{P}_{n+1}$, leading to the somewhat surprising conclusion that although $\mathfrak{P}_n$ tends to infinity with $n$, the inequality $\mathfrak{P}_n>\mathfrak{P}_{n+1}$ is more frequent than its reverse.

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