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.

Riff 2023

Riff 2023

Rote 2023

Rote 2023


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