Riffs and Rotes • Happy New Year 2026

[Jon Awbrey]

Riffs and Rotes • Happy New Year 2026
• https://inquiryintoinquiry.com/2026/01/01/riffs-and-rotes-happy-new-year-2026/

All,

There's a deep, mathematical significance I see in the following structures,
and I'm hoping one day to find a way to explain everything that I see there.
Meanwhile, you may take them as an amusing diversion in recreational maths.

Let p_n = the n-th prime. Then we have the following development.

2026 = 2 ∙ 1013

= p_1 p_170

= p_1 p_{2 ∙ 5 ∙ 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}}}}

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
• https://inquiryintoinquiry.com/wp-content/uploads/2026/01/riff-2026-card.png

Rote 2026
• https://inquiryintoinquiry.com/wp-content/uploads/2026/01/rote-2026-card.png

Reference —

Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotes

Regards,

Jon

cc: https://www.academia.edu/community/VBA6Qz
cc: https://www.researchgate.net/post/Riffs_and_Rotes_Happy_New_Year_2026