论文标题
指数元序列
Exponential prime sequences
论文作者
论文摘要
形式的$ \ lfloor a c^{n^d}+b \ rfloor $，$ n \ geq 0 $的无限指数序列被证明存在，对于精心选择的真实常数$ a> 0 $，$ b $，$ b $，$ c> 1 $，$ d> 1 $，$ d> 1 $，假设Prime on Prime ppirite ppirite lipe lip lip ppirite lise lip lip ppirite lip lip lip lip lip。有这样的序列的无穷大。可能增长率最小的序列特别感兴趣。这项工作的重点是$ a = 1 $，$ b \ in \ {0,1 \} $，其具有最小的常数$ c $给定$ d> 1 $，并且具有最小的$ d $，给定$ c = 2 $。特别是，我们证明了四个无限指数prime序列的存在$ u_0（n）= \ lfloor c_0^{n \ sqrt {n}} \ rfloor $，$ n \ geq 1 $，带有$ C_0 = 2.0077340803 ... $，$，$，$，$ u_1（n） C_1^{n \ sqrt {n}} \ rfloor $，$ n \ geq 0 $，带有$ C_1 = 2.2679962677 ... $，$，$ v_0（n）= \ lfloor 2^{n^{n^{d_0}}} $ d_0 = 1.5039285240 ... $，$ v_1（n）= 1+\ lfloor 2^{n^{d_1}} \ rfloor $，$ n \ geq 0 $，带有$ d_1 = 1.7355149500 ... $。
Infinite exponential sequences of distinct prime numbers of the form $\lfloor a c^{n^d}+b\rfloor$, $n\geq 0$, are proved to exist for well chosen real constants $a>0$, $b$, $c>1$, $d>1$, assuming Cramer's conjecture on prime gaps. There is an infinity of such prime sequences. Sequences having the least possible growth rate are of particular interest. This work's focus is on prime sequences with $a=1$, $b \in \{0,1\}$, that have the smallest possible constant $c$ given $d>1$, and sequences with the smallest possible $d$, given $c=2$. In particular, we prove the existence of the four infinite exponential prime sequences $u_0(n)=\lfloor c_0^{n\sqrt{n}}\rfloor$, $n\geq 1$, with $c_0=2.0073340803...$, $u_1(n)=1+\lfloor c_1^{n\sqrt{n}}\rfloor$, $n\geq 0$, with $c_1=2.2679962677...$, $v_0(n)=\lfloor 2^{n^{d_0}}\rfloor$, $n\geq 1$, with $d_0=1.5039285240...$, and $v_1(n)=1+\lfloor 2^{n^{d_1}}\rfloor$, $n\geq 0$, with $d_1=1.7355149500...$.
