论文标题
转移指数的复合值
Composite values of shifted exponentials
论文作者
论文摘要
一个众所周知的开放问题要求表明$ 2^n+5 $对于几乎所有$ n $的值都是综合的。这是吉尔·卡莱（Gil Kalai）提出的可能是一个可能的多层项目，并最初由克里斯托弗·霍利（Christopher Hooley）提出。我们表明，假设GRH和一对相关性猜想的一种形式，此问题的答案是肯定的。实际上，我们不需要对配对相关性的全部力量，并且足以在其暗示的Chebotarev密度定理中假设Brun-titchmarsh不等式的概括。我们的方法适用于表单$ a^n-b $的任何移动指数序列，并表明，在相同的假设下，此类数字是$ k $ - 对于密度$ 0 $ 0 $的自然数量$ n $的数字。此外，我们表明，每当$（a，b）\ neq（2，1）$时，几乎所有Primes $ p $ $ a^p-b $都是合成的。
A well-known open problem asks to show that $2^n+5$ is composite for almost all values of $n$. This was proposed by Gil Kalai as a possible Polymath project, and was posed originally by Christopher Hooley. We show that, assuming GRH and a form of the pair correlation conjecture, the answer to this problem is affirmative. We in fact do not need the full power of the pair correlation conjecture, and it suffices to assume a generalization of the Brun-Titchmarsh inequality for the Chebotarev density theorem that is implied by it. Our methods apply to any shifted exponential sequence of the form $a^n-b$ and show that, under the same assumptions, such numbers are $k$-almost primes for a density $0$ of natural numbers $n$. Furthermore, we show that $a^p-b$ is composite for almost all primes $p$ whenever $(a, b) \neq (2, 1)$.
