Lmst

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

Performance and Numerical Aspects of Decompositional Factorizations with FP64 Floating-Point Emulation in INT8

#NVIDIA #Int8 #FP64 #Factorization

https://hgpu.org/?p=30278

More #quantum computer #factorization fun:

https://www.johndcook.com/blog/2025/09/28/extrapolating-quantum-factoring/

If we extrapolate from the two points we have, factoring '15' in 2001 and '21' (kind of) in 2012, and would like to be able to factorize a 1024 bit number in 2035, we should be able to factorize 40 bit numbers now. Which we don't.

The author proposes to add new data points as they come in. I'm looking forward to that :)

This is excellent:

Replication of Quantum Factorisation Records with an 8-bit Home Computer, an Abacus, and a Dog

https://eprint.iacr.org/2025/1237.pdf

Selected hilarious quotations:

We use the UK form “factorise” here in place of the US variants “factorize” or “factor” in order to avoid the 40% tariff on the US term.

Similarly, we refer to an abacus as “an abacus” rather than a digital computer, despite the fact that it relies on digital manipulation to effect its computations.

The VIC-20 was a popular home computer in the 1980s that used the then popular 6502 microprocessor from 1975. Since this processor uses transistors, and transistors work by using quantum effects, a 6502 is as much a quantum device as is a D-Wave “quantum computer”

#vic20 #quantumComputing #factorization #factorisation

Harnessing Batched BLAS/LAPACK Kernels on GPUs for Parallel Solutions of Block Tridiagonal Systems

#ROCm #CUDA #BLAS #Factorization #Package

https://hgpu.org/?p=30168

LOL

"In this paper we showed how to replicate current quantum factorisation records using first a VIC-20 8-bit home computer from 1981, then an abacus, and finally a dog....[W]e rank a VIC-20 above an abacus, an abacus above a dog, and a dog above a quantum factorisation physics experiment. Finally, we provided standard evaluation criteria for future claimed quantum factorisations."

#physics #vic20 #factorization #quantumbullshit #quantumcomputing #funny #dogs #abacus

https://eprint.iacr.org/2025/1237.pdf

> Researchers claim to have used a #quantumComputer to factor a 2,048-bit #RSA integer.

> But the RSA number evaluated was the product of two prime factors that were too close together.

> As with a parlor magician's card deck that's been stacked for a card trick

> #Quantum #factorization is performed using sleight-of-hand numbers that have been selected to make them very easy to factorize using a #physics experiment

https://www.theregister.com/2025/07/17/quantum_cryptanalysis_criticism/

#quantumComputing #computerScience #cryptography

Researchers Use 1981 Home Computer to Match #Quantum #Factorization Records, Exposing Limitations

https://biggo.com/news/202507131329_Home_Computer_Matches_Quantum_Factorization_Records

One day, one decomposition
A000028: Let k = p_1^e_1 p_2^e_2 p_3^e_3 ... be the prime factorization of n. Sequence gives k such that the sum of the numbers of 1's in the binary expansions of e_1, e_2, e_3, ... is odd.

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A000028.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A000028.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A000028.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #primes #factorization #PrimeNumbers #binary #expansions #graph #threejs #webGL

Decomposition into weight × level + jump of A000028 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Decomposition into weight × level + jump of A000028 in 2D (log(weight), log(level))

One day, one decomposition
A276078: Numbers n in whose prime factorization no exponent of any prime(k) exceeds k

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A276078.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A276078.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A276078.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #factorization #exponent #primes #PrimeNumbers #graph #threejs #webGL

Decomposition into weight × level + jump of A276078 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A276078 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

One day, one decomposition
A260047: Composites whose prime factorization in base 3 is an anagram of the number in base 3

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A260047.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A260047.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A260047.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #prime #factorization #anagram #graph #threejs #webGL

Decomposition into weight × level + jump of A260047 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A260047 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

One day, one decomposition
A260046: Composites whose prime factorization in base 2 is an anagram of the number in base 2

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A260046.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A260046.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A260046.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #composites #prime #factorization #anagram #graph #threejs #webGL

Decomposition into weight × level + jump of A260046 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A260046 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

A first successful factorization of RSA-2048 integer by D-Wave quantum computer

https://www.sciopen.com/article/10.26599/TST.2024.9010028

#HackerNews #quantumcomputing #RSA2048 #DWave #factorization #cybersecurity #technews

One day, one decomposition
A242414: Numbers whose prime factorization viewed as a tuple of nonzero powers is palindromic

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A242414.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A242414.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #prime #factorization #palindromic #graph #threejs #webGL

Decomposition into weight × level + jump of A242414 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A242414 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

One day, one decomposition
A212168: Numbers n such that the maximal exponent in its prime factorization is less than the number of positive exponents (A051903(n) < A001221(n))

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A212168.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A212168.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #exponent #prime #factorization #PrimeNumbers #graph #threejs #webGL

Decomposition into weight × level + jump of A212168 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A212168 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

One day, one decomposition
A212166: Numbers k such that the maximum exponent in its prime factorization equals the number of positive exponents (A051903(k) = A001221(k))

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A212166.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A212166.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #exponent #prime #factorization #graph #threejs #webGL

Decomposition into weight × level + jump of A212166 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A212166 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

One day, one decomposition
A212165: Numbers k such that the maximum exponent in its prime factorization is not less than the number of positive exponents (A051903(k) >= A001221(k))

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A212165.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A212165.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #exponent #prime #factorization #graph #threejs #webGL

Decomposition into weight × level + jump of A212165 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A212165 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

One day, one decomposition
A212164: Numbers k such that the maximum exponent in its prime factorization is greater than the number of positive exponents (A051903(k) > A001221(k))

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A212164.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A212164.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #exponent #prime #factorization #graph #threejs #webGL

Decomposition into weight × level + jump of A212164 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A212164 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

'Guaranteed Nonconvex Factorization Approach for Tensor Train Recovery', by Zhen Qin, Michael B. Wakin, Zhihui Zhu.

http://jmlr.org/papers/v25/24-0029.html

#tensor #tensors #factorization

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

#Factorization

Client Info