Exponential prime sequences

arXiv2011.14653MaRDI QIDQ6354846

Bernard Montaron

Publication date: 30 November 2020

Abstract: Infinite exponential sequences of distinct prime numbers of the form $lfloora{c}^{n^d}+bfloor$, $ngeq0$, 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 $a=1$, $bin0,1$, 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 $u_0(n)=lfloor{c}_0^{nsqrtn}floor$, $ngeq1$, with $c_0=2.0073340803...$, $u_1(n)=1+lfloor{c}_1^{nsqrtn}floor$, $ngeq0$, with $c_1=2.2679962677...$, $v_0(n)=lfloor{2}^{n^{d_0}}floor$, $ngeq1$, with $d_0=1.5039285240...$, and $v_1(n)=1+lfloor{2}^{n^{d_1}}floor$, $ngeq0$, with $d_1=1.7355149500...$.

This page was built for publication: Exponential prime sequences

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6354846)