Riffs and Rotes • Happy New Year 2023

$\text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}.$

$\text{Then} ~ 2023 = 7 \cdot {17}^2 = p_{4} p_{7}^{2} = p_{{p_1}^{p_1}} p_{{p_4}}^{p_1} = p_{{p_1}^{p_1}} p_{{p_{{p_1}^{p_1}}}}^{p_1}$

No information is lost by dropping the terminal 1s.  Thus we may write the following form.

$2023 = p_{p^p} p_{{p_{p^p}}}^p$

The article referenced below tells how forms like these correspond to a family of digraphs called riffs and a family of graphs called rotes.  The riff and rote for $2023$ are shown in the next two displays.

Reference

This entry was posted in Algebra, Combinatorics, Graph Theory, Group Theory, Logic, Mathematics, Number Theory, Riffs and Rotes and tagged , , , , , , , . Bookmark the permalink.

This site uses Akismet to reduce spam. Learn how your comment data is processed.