Riffs and Rotes • Happy New Year 2021

[Jon Awbrey]

Cf: Riffs and Rotes • Happy New Year 2021
http://inquiryintoinquiry.com/2021/01/03/riffs-and-rotes-happy-new-year-2021/

All,

Apart from their abstract beauty, Riffs and Rotes are structures
I discovered while playing around with Gödel numberings of graphs
and digraphs so they have a deep connection with the mathematical
infrastructure of logic. At any rate, for your musement, here are
the Riff and Rote for 2021.

Letting p_n = pₙ = the nth prime, beginning 2, 3, 5, 7, 11, 13, etc.

2021 = 43 × 47 = p₁₄ × p₁₅ = p₂ₓ₇ × p₃ₓ₅

2021 = 43 × 47 = p_14 × p_15 = p_{p_1 p_4} × p_{p_2 p_3} = et sic deinceps ...

Riff 2021
https://inquiryintoinquiry.files.wordpress.com/2021/01/riff-2021-big-2.0.jpg

Rote 2021
https://inquiryintoinquiry.files.wordpress.com/2021/01/rote-2021-big-2.0.jpg

Reference
=========

Riffs and Rotes ( https://oeis.org/wiki/Riffs_and_Rotes )

Happy New Year !!!

Jon

inquiry into inquiry: https://inquiryintoinquiry.com/
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[Jon Awbrey]

Cf: Riffs and Rotes • 1
https://inquiryintoinquiry.com/2013/01/28/riffs-and-rotes-1/

Re: Richard J. Lipton • Making Primes More Random
https://rjlipton.wpcomstaging.com/2013/01/26/making-primes-more-random/

All,

There's a study called “generalized primes” which investigates in a more
general way the relationship between arbitrary elements called “primes”
and the “composites” which can be formed from them according to specified
rules of composition. Comparisons can be made among a variety numerical
systems or any orders of combinatorial species one might imagine.
I seem to recall at least one old monograph by Rademacher on the subject.

My fascination with questions like that led me many years ago to the
“Riff and Rote” trick, a special case of the “Make A Picture” trick.
There's a bit on that in the following article.

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

Regards,

Jon

[Jon Awbrey]

Cf: Riffs and Rotes • 2
https://inquiryintoinquiry.com/2013/03/03/riffs-and-rotes-2/

Re: Peter Cameron • Addition and Multiplication of Natural Numbers
https://cameroncounts.wordpress.com/2013/03/03/two-lectures/

All,

The interaction between addition and multiplication in the natural numbers
has long been an interest of mine, leading to broader questions about the
relationship between algebra and combinatorics. My gropings with these
enigmas led me to the structures of Riffs and Rotes, extracting what
we might think of as the “purely combinatorial” properties of primes
factorizations. Thinking of the additive structure of the positive
integers as embodied in their total linear ordering, the following
two questions arise.

• How much of the natural ordering of the natural numbers
is purely combinatorial?

• What additional axioms on the partial orders of Riffs and Rotes
would restore their natural order?

Reference
=========

[Jon Awbrey]

Cf: Riffs and Rotes • 59281
https://inquiryintoinquiry.com/2021/09/05/riffs-and-rotes-59281/

Re: Persiflage ( https://www.galoisrepresentations.com/about/ )
::: 59281 ( https://www.galoisrepresentations.com/2021/08/19/59281/ )

<QUOTE Persiflage:>
Numberfile • What's Special About 59,281?
( https://www.youtube.com/watch?v=Umdv0GmO73g )

If p is prime then the decimal expansion of 1/p repeats, so
it makes sense to talk about the “average” of the digits of 1/p.
The average can be bigger than 4.5, equal to 4.5, or less than 4.5.
Which is most likely? Which is least likely? Click to find out.

Challenge Problem. Is there any prime for which
the digit average is bigger than it is for p = 59,281?
</QUOTE>

I can’t imagine this will help with the problem,
it’s just a thing I do with interesting numbers
I encounter …

See Riffs and Rotes ( https://oeis.org/wiki/Riffs_and_Rotes )
for the basic idea.

Figure 1. Doubly Recursive Factorization 59281
https://inquiryintoinquiry.files.wordpress.com/2021/09/doubly-recursive-factorization-59281.png

Here is the Riff for 59281

Figure 2. Riff 59281
https://inquiryintoinquiry.files.wordpress.com/2021/08/riff-59281.png

Here is the Rote for 59281

Figure 3. Rote 59281
https://inquiryintoinquiry.files.wordpress.com/2021/08/rote-59281.png

One peculiar property of this number I notice is its being
“square-free all the way down”. Once again, I have no clue
whether that has anything to do with the problem at hand.

Regards,

Jon