#number-theory — Public Fediverse posts

The Method Matters More Than the Number: A Proposed Improvement to the Guth–Maynard Bound 30/13

The interest in lowering the coefficient to 24000/10451 ≈ 2.29643 lies mainly in the technique. The method aims to preserve relationships between oscillating terms, including how they can cancel, as sums are transformed. Bounding terms by their sizes alone can erase that information and weaken the estimates. Preserving it might help with other problems in number theory and harmonic analysis that share similar structures. That potential remains to be explored; the paper’s result is still pending verification.

The entire process was interactive and iterative: hundreds of exchanges with a frontier model, without a single autonomous agent, without using the API, and with zero client-side automation. My local computing resources were ordinary home equipment. I proposed, questioned and redirected each step. I believe this continuous involvement was essential to bring human creativity into the process and guide the model’s reasoning. Many conversations bordered on the absurd, the unfamiliar or the unorthodox. For me, that is where the human–machine “magic” emerged.

The manuscript is currently almost indigestible for a human reader: it contains failed attempts, detours and corrections. I estimate that the proof’s chain of arguments could fit into a dozen pages, out of the current 172. That distillation, and of course, formal proof, have yet to be done. Finally, math community help will ease human learning.

Apologies for the AI-generated image: it may contain errors.

"A Zero-Density Estimate with Coefficient 24000/10451 < 30/13 for the Riemann Zeta-Function"
https://doi.org/10.13140/RG.2.2.27464.07682
#numbertheory #ZeroDensity

Fable 5.1 - max: pedagogical report for Graduate / researcher new to the #decompwlj ➡️ https://decompwlj.com/WeightLevelJump_Primer.html

How to read this report

Primes VS Model A, the classic Cramér sequence

Twelve fingerprints

The complete status ledger

#math #NumberTheory #number #sequence #AI #Claude #Fable #Anthropic

Fable 5.1: pedagogical report for Graduate / researcher new to the #decompwlj ➡️ https://decompwlj.com/WeightLevelJump_Primer.html

How to read this report

Primes VS Model A, the classic Cramér sequence

Twelve fingerprints

The complete status ledger

#math #NumberTheory #number #sequence #AI #Claude #Fable #Anthropic

Fable 5.1: pedagogical report for Graduate / researcher new to the #decompwlj ➡️ https://decompwlj.com/WeightLevelJump_Primer.html

How to read this report

Primes VS Model A, the classic Cramér sequence

Twelve fingerprints

The complete status ledger

#math #NumberTheory #AI #Claude #Fable #graph #sequence

#decompwlj #math #mathematics #NumberTheory #numbers #sequence #OEIS #JavaScript #threejs #3D #graph #decomposition #classification #sieve

A081605 ➡️ https://oeis.org/A081605

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A081605.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A081605.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A081605.html

#decompwlj #math #numbers #sequence #threejs #3D #graph

🚨 BREAKING NEWS 🚨

Number theorist Zhi-Wei Sun, with AI assistance, has published a paper on arXiv claiming to prove that Catalan’s constant is irrational.

If the result holds up to scrutiny, this would be a major breakthrough in number theory. The irrationality/transcendence of Catalan’s constant has long been an important open problem.

Preprint link: https://arxiv.org/pdf/2609.04176

#GPT #NumberTheory #AI4Maths #CatalanConstant #ChatGPT #AIinMaths

Alright, future engineers!
**Modular Arithmetic:** Numbers wrap around after a certain value (the modulus).
Ex: `(10 + 4) mod 12 = 14 mod 12 = 2`. Think of a clock!
Pro-Tip: It's super useful in cryptography & computer science for cycles/remainders!
#NumberTheory #DiscreteMath #STEM #StudyNotes

4/4: The base-10 palindromes - a "base" sequence. They have often weird/fractal graphs

#decompwlj #math #Anthropic #Claude #Opus5 #sequence #numbers #NumberTheory

3/4: The triangular numbers are all level-classified (trivial)

#decompwlj #math #Anthropic #Claude #Opus5 #sequence #numbers #NumberTheory

2/4: All 5,761,452 decomposable primes < 10^8

#decompwlj #math #Anthropic #Claude #Opus5 #sequence #numbers #NumberTheory

Opus 5 - High-res PNG (slight origin offset for levels 1) 🧵

1/4: The base case: the natural numbers - the sieve of Eratosthenes

#decompwlj #math #Anthropic #Claude #Opus5 #sequence #numbers #NumberTheory

Alright, future engineers!
**Modulo Arithmetic:** Finding the remainder after division.
Ex: `17 mod 5 = 2`.
Pro-Tip: Crucial for cryptography, clock arithmetic, and hashing!
#DiscreteMath #NumberTheory #STEM #StudyNotes

Opus 5 - High-res PNG (slight offset from the origin):
The base-10 palindromes - a plate that repeats itself every two digits

#decompwlj #math #graph #AI #Anthropic #Claude #Opus #Opus5 #mathematics #sequence #base #numbers #NumberTheory #sieve #arithmetic #classification #decomposition

Opus 5 - High-res PNG (slight offset from the origin):
The triangular numbers are all level-classified

#decompwlj #math #graph #AI #Anthropic #Claude #Opus #Opus5 #mathematics #sequence #triangular #numbers #TriangularNumbers #NumberTheory #sieve #arithmetic #classification #decomposition

Opus 5 - High-res PNG (slight offset from the origin):
The base case: the natural numbers - the weight sheet is the sieve of Eratosthenes

#decompwlj #math #graph #AI #Anthropic #Claude #Opus #Opus5 #mathematics #sequence #natural #numbers #NaturalNumbers #sieveOfEratosthenes #NumberTheory #sieve #arithmetic #classification #decomposition

Opus 5 - High-res PNG (slight offset from the origin):
All 5,761,452 decomposable primes < 10^8

#decompwlj #math #graph #AI #Anthropic #Claude #Opus #Opus5 #mathematics #sequence #numbers #prime #PrimeNumbers #NumberTheory #sieve #arithmetic #classification #decomposition

Opus 5 Max: A pedagogical treatise for Graduate / researcher new to the decomposition into weight × level + jump ➡️ https://decompwlj.com/Opus5_decompwlj_pedagogical_treatise_9th_edition_2026_08_16.html

The weight is unique:
"The weight is unique. Minimality pins it. There is no choice of parameter anywhere in Definition 1 — no window, no threshold, no cut-off. This is what makes the construction worth studying at all: it is a canonical map, and the objects it produces are invariants of the sequence."

Proof of Conjecture 9:
"This is the framework's native question: it cannot even be asked without the coordinates, and it has an answer."

The atlas as an instrument:
"About a thousand OEIS sequences have been decomposed and published at https://decompwlj.com. Today that is an archive. It could be an instrument."

#decompwlj #math #AI #Anthropic #Claude #Opus #Opus5 #mathematics #OEIS #sequence #numbers #PrimeNumbers #NumberTheory #sieve #arithmetic #threejs #research #conjecture #graph #sieveOfEratosthenes #FundamentalTheoremOfArithmetic #classification #decomposition

A072502: Numbers that are run sums (trapezoidal, the difference between two triangular numbers) in exactly 3 ways
A072502 ➡️ https://oeis.org/A072502

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A072502.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A072502.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A072502.html

#decompwlj #math #mathematics #maths #sequence #OEIS #JavaScript #graph #3D #threejs #webGL #triangular #numbers #primes #PrimeNumbers #palindromes #animation #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research

My data have not been verified but my work is highly reproducible.
- Downloads (csv, img, dump) ➡️ https://decompwlj.com/
- Algorithms ➡️ https://oeis.org/wiki/Decomposition_into_weight_*_level_%2B_jump#Algorithms

#decompwlj #math #mathematics #maths #sequence #OEIS #PARIGP #Downloads #Algorithms #numbers #primes #PrimeNumbers #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research

A072437: Numbers with no prime factors of form 4*k+3
A072437 ➡️ https://oeis.org/A072437

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A072437.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A072437.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A072437.html

#decompwlj #math #mathematics #maths #sequence #OEIS #JavaScript #graph #3D #threejs #webGL #triangular #numbers #primes #PrimeNumbers #palindromes #animation #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research

Opus 5 Max: A pedagogical treatise for Graduate / researcher new to the decomposition into weight × level + jump ➡️ https://decompwlj.com/Opus5_decompwlj_pedagogical_treatise_9th_edition_2026_08_16.html

The weight is unique:
"The weight is unique. Minimality pins it. There is no choice of parameter anywhere in Definition 1 — no window, no threshold, no cut-off. This is what makes the construction worth studying at all: it is a canonical map, and the objects it produces are invariants of the sequence."

Proof of Conjecture 9:
"This is the framework's native question: it cannot even be asked without the coordinates, and it has an answer."

The atlas as an instrument:
"About a thousand OEIS sequences have been decomposed and published at https://decompwlj.com. Today that is an archive. It could be an instrument."

#decompwlj #math #AI #Anthropic #Claude #Opus #Opus5 #mathematics #OEIS #sequence #numbers #PrimeNumbers #NumberTheory #sieve #arithmetic #threejs #research #conjecture #graph #sieveOfEratosthenes #FundamentalTheoremOfArithmetic #classification #decomposition

A072225: Numbers k such that prime(k) + prime(k+1) + prime(k+2) is prime
A072225 ➡️ https://oeis.org/A072225

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A072225.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A072225.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A072225.html

#decompwlj #math #mathematics #maths #sequence #OEIS #JavaScript #graph #3D #threejs #webGL #triangular #numbers #primes #PrimeNumbers #palindromes #animation #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research

In the decomposition into weight × level + jump, the Twin Prime Conjecture becomes: "The number of prime numbers with a weight of 3 is infinite"

#decompwlj #math #mathematics #maths #sequence #OEIS #JavaScript #graph #3D #threejs #webGL #numbers #TwinPrimes #twin #primes #conjecture #PrimeNumbers #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research

A072202: Same numbers of prime factors of forms 4*k+1 and 4*k+3, counted with multiplicity
A072202 ➡️ https://oeis.org/A072202

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A072202.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A072202.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A072202.html

#decompwlj #math #mathematics #maths #sequence #OEIS #JavaScript #graph #3D #threejs #webGL #triangular #numbers #primes #PrimeNumbers #palindromes #animation #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research

Opus 5 Max: A pedagogical treatise for Graduate / researcher new to the decomposition into weight × level + jump ➡️ https://decompwlj.com/Opus5_decompwlj_pedagogical_treatise_9th_edition_2026_08_16.html

The weight is unique:
"The weight is unique. Minimality pins it. There is no choice of parameter anywhere in Definition 1 — no window, no threshold, no cut-off. This is what makes the construction worth studying at all: it is a canonical map, and the objects it produces are invariants of the sequence."

Proof of Conjecture 9:
"This is the framework's native question: it cannot even be asked without the coordinates, and it has an answer."

The atlas as an instrument:
"About a thousand OEIS sequences have been decomposed and published at https://decompwlj.com. Today that is an archive. It could be an instrument."

#decompwlj #math #AI #Anthropic #Claude #Opus #Opus5 #mathematics #OEIS #sequence #numbers #PrimeNumbers #NumberTheory #sieve #arithmetic #threejs #research #conjecture #graph #sieveOfEratosthenes #FundamentalTheoremOfArithmetic #classification #decomposition

A072140: The period length of the 'Reverse and Subtract' trajectory of n is greater than 1
A072140 ➡️ https://oeis.org/A072140

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A072140.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A072140.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A072140.html

#decompwlj #math #mathematics #maths #sequence #OEIS #JavaScript #graph #3D #threejs #webGL #triangular #numbers #primes #PrimeNumbers #palindromes #animation #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research

We see the fundamental theorem of arithmetic and the sieve of Eratosthenes in the decomposition into weight × level + jump of natural numbers. Applied to prime numbers this decomposition leads to a new classification of primes.

#decompwlj #math #mathematics #maths #sequence #OEIS #graph #numbers #primes #PrimeNumbers #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #SieveofEratosthenes #sieve #Eratosthenes #fundamental #theorem #arithmetic #research

A072055: a(n) = 2*prime(n)+1
A072055 ➡️ https://oeis.org/A072055

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A072055.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A072055.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A072055.html

#decompwlj #math #mathematics #maths #sequence #OEIS #JavaScript #graph #3D #threejs #webGL #triangular #numbers #primes #PrimeNumbers #palindromes #animation #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research

Opus 5 Max: A pedagogical treatise for Graduate / researcher new to the decomposition into weight × level + jump ➡️ https://decompwlj.com/Opus5_decompwlj_pedagogical_treatise_9th_edition_2026_08_16.html

The weight is unique:
"The weight is unique. Minimality pins it. There is no choice of parameter anywhere in Definition 1 — no window, no threshold, no cut-off. This is what makes the construction worth studying at all: it is a canonical map, and the objects it produces are invariants of the sequence."

Proof of Conjecture 9:
"This is the framework's native question: it cannot even be asked without the coordinates, and it has an answer."

The atlas as an instrument:
"About a thousand OEIS sequences have been decomposed and published at https://decompwlj.com. Today that is an archive. It could be an instrument."

#decompwlj #math #AI #Anthropic #Claude #Opus #Opus5 #mathematics #OEIS #sequence #numbers #PrimeNumbers #NumberTheory #sieve #arithmetic #threejs #research #conjecture #graph #sieveOfEratosthenes #FundamentalTheoremOfArithmetic #classification #decomposition