#numbertheory — Public Fediverse posts

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 - 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

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 - 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 - 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 - 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 - 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 #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 #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

Number theory was the one area of mathematics on which Pierre de Fermat (1607–65) worked throughout his life, and he found it ‘very beautiful and very subtle’.

Among other results, he said that the Polygonal Number Theorem (which asserts that every natural number is the sum of at most $n$ $n$-gonal numbers) was ‘a most beautiful and wholly general proposition […] this marvellous proposition’.

(He offered no proof of this result, but claimed to have one in a marginal note to Diophantus' Arithmetica; this was the same book in which he noted what became known as Fermat's Last Theorem.)

Fermat also seems to have counted magic squares and analogous configurations as part of number theory, and wrote that: ‘I know hardly anything more beautiful in arithmetic than these numbers that some call planetary and others magic’. (The term ‘planetary’ is derived from certain treatises linking the magic squares to planets used in talismans.)

He said he had found a rule to find magic cubes (one of his examples is in the attached image) and also determined how many different ways each such cube can be arranged, which he called ‘one of the most beautiful things in arithmetic’.

1/2

[Each day of February, I am posting a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.]

#Fermat #NumberTheory #HistMath #MathematicalBeauty #MagicSquare

On the way to #Integers2025 , I told my colleague, trapped in the car with me, that I was going to be lost in the #NumberTheory talks. I mused that all I remembered from abstract algebra that might be relevant was Cayley groups.

I didn't see that name (Cayley) until day three. I failed even at knowing what I wouldn't know.

The second day of #Integers2025 was excellent! I heard some great talks, especially one from Carrie Finch-Smith. Me and the other three gamesters I know about here also took some time and I think we found a game to be #PSPACE complete, so we're definitely going to be writing that up!

Tonight we got to see a tree that owns itself. Cool stuff!

The organizers have been totally awesome. Great conference so far!

#NumberTheory #CombinatorialGames

Day 1 of #Integers2025 just wrapped up! Integers is a #NumberTheory and #Combinatorics conference. There's historically a bit of CGT too. Here are my summaries of the #CombinatorialGames talks: https://combinatorialgametheory.blogspot.com/2025/05/integers-2025-cgt-talks.html

Two of the talks were the result of #UndergraduateResearch!

I just uploaded some notes (in French) for my last Friday “Séminaire Bourbaki du vendredi” talk on the Weil conjectures (history, approaches, generalizations): I recounted the history of the conjectures by Weil on the number of solutions of polynomial equations in finite fields, and some of the approaches that have been proposed to solve them.

https://webusers.imj-prg.fr/~antoine.chambert-loir/publications/exposes/weil-20221118.pdf

#WeilConjectures #NumberTheory #Bourbaki