Riffs and Rotes • Happy New Year 2025

[Jon Awbrey]

Riffs and Rotes • Happy New Year 2025
• https://inquiryintoinquiry.com/2025/01/01/riffs-and-rotes-happy-new-year-2025/

Let pₙ = the n-th prime.

Then 2025

= 81 ∙ 25

= 3⁴ ∙ 5²

= (p₂)⁴ ∙ (p₃)²

= {p_2}^4 ∙ {p_3}^2

= {p_2}^{{p_1}^{p_1}} ∙ {p_3}^{p_1}

= {p_{p_1}}^{{p_1}^{p_1}} ∙ {p_{p_2}}^{p_1}

= {p_{p_1}}^{{p_1}^{p_1}} ∙ {p_{p_{p_1}}}^{p_1}

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

2025 = {p_p}^{p^p} ∙ {p_{p_p}}^p

The article linked below tells how forms like this correspond
to a family of digraphs called “riffs” and a family of graphs
called “rotes”. The riff and rote for 2025 are shown in the
next two Figures.

Riff 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/riff-2025.png

Rote 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/rote-2025.png

Reference —

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

Regards,

Jon

cc: https://www.academia.edu/community/lnYzY3
cc: https://www.researchgate.net/post/Riffs_and_Rotes_Happy_New_Year_2025