Riffs and Rotes • Happy New Year 2026

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

\begin{array}{llcl}  \text{Then} & 2026 & = & 2 \cdot 1013  \\  && = & p_1 p_{170}  \\  && = & p_1 p_{2 \cdot 5 \cdot 17}  \\  && = & p_1 p_{p_1 p_3 p_7}  \\  && = & p_1 p_{p_1 p_{p_2} p_{p_4}}  \\  && = & p_1 p_{p_1 p_{p_{p_1}} p_{p_{{p_1}^{p_1}}}}  \end{array}

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

2026 = p p_{p p_{p_p} p_{p_{p^p}}}

The article linked below tells how forms of that order correspond to a family of digraphs called riffs and a family of graphs called rotes.  The riff and rote for 2026 are shown in the next two Figures.

Riff 2026

Riff 2026

Rote 2026

Rote 2026

Reference

cc: Academia.eduLaws of FormMathstodonResearch Gate
cc: CyberneticsStructural ModelingSystems ScienceSyscoi

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