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

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

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

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

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
• 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

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

#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes

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

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

\( \text{Then} ~ 2025
= 81 \cdot 25
= 3^4 5^2 \)

\( = {p_2}^4 {p_3}^2
= {p_2}^{{p_1}^{p_1}} {p_3}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_2}}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_{p_1}}}^{p_1} \)

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 of that sort 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

#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes

Riffs and Rotes • Happy New Year 2024
• https://inquiryintoinquiry.com/2024/01/02/riffs-and-rotes-happy-new-year-2024/

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

\( \text{Then} ~ 2024
= 8 \cdot 11 \cdot 23
= p_{1}^{3} p_{5} p_{9}
= p_{1}^{p_2} p_{p_3} p_{p_2^2}
= p_{1}^{p_{p_1}} p_{p_{p_2}} p_{p_{p_1}^{p_1}}
= p_{1}^{p_{p_1}} p_{p_{p_{p_1}}} p_{p_{p_1}^{p_1}} \)

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

\[ 2024 = p^{p_p} \cdot p_{p_{p_p}} \cdot p_{p_p^p} \]

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 2024 are shown in the next two Figures.

Riff 2024
• https://inquiryintoinquiry.files.wordpress.com/2024/01/riff-2024.png

Rote 2024
• https://inquiryintoinquiry.files.wordpress.com/2024/01/rote-2024.png

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

#GraphTheory #NumberTheory #Primes #PrimeNumbers #RiffsAndRotes

#TodaysAcronym ☞ #DIYFFAN

#DoItYourselfFunFactoidsAboutNumbers

#RiffsAndRotes
• https://oeis.org/wiki/Riffs_and_Rotes

Riffs and Rotes • Happy New Year 2023
• https://inquiryintoinquiry.com/2023/01/01/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.

\[ 2023=p_{p^p} p_{p_{p^p}}^p \]

Forms like these correspond to a family of #Digraphs called #Riffs and a family of #Graphs called #Rotes.

#GraphTheory #NumberTheory #Primes #PrimeNumbers #RiffsAndRotes

#RiffsAndRotes • Happy New Year 2023
• https://inquiryintoinquiry.com/2023/01/01/riffs-and-rotes-happy-new-year-2023/

\(2023 = 7 \cdot {17}^2 = p_4 p_7^2 = p_{p_1^{p_1}} p_{p_4}^{p_1} = p_{p_1^{p_1}} p_{p_{p_1^{p_1}}}^{p_1}\)

Happy New Year 🎉

\(2023 = p_4^1 p_7^2 = p_{p_1^2} p_{p_4^1}^2\)

#RiffsAndRotes
• https://oeis.org/wiki/Riffs_and_Rotes

Just experimenting a little —
will add more explanation later —

#RiffsAndRotes
• https://oeis.org/wiki/Riffs_and_Rotes

#Riffs from 1 to 60
• https://oeis.org/w/images/1/17/Animation_Riff_60_x_0.16.gif

#Rotes from 1 to 60
• https://oeis.org/w/images/e/ee/Animation_Rote_60_x_0.16.gif

#Arithmetization #GödelNumbering
#DoublyRecursiveFactorization #DRF
#Arithmetic #GraphTheory #NumberTheory
#Primes #PrimeNumbers #PrimesFactorization