#numbertheory

Saw this statement on another site: "Any prime number higher than three, when squared and subtracted by one, will always turn out to be a multiple of 24."

I never heard of this one before. Is there a proof of this property? Definitely needs a bit more investigation.

#PrimeNumber #proof #properties #primes #NumberTheory

is anyone from mathstodon willing to critique this preprint? the author claims to have solved the Collatz conjecture and I have not found any holes in the proof yet.

https://www.preprints.org/manuscript/202502.2072/v3

The author first shows that collatz cycles have bounded growth, then proves the uniqueness of the 4-2-1 cycle. The ensuing claim is that this implies that all paths converge to 4-2-1.

It's all quite easy for me to follow, which raises my skepticism a bit.

#collatz #numbertheory

One of the best things I saw this week: a paper uncovering alien signals in the Riemann Zeta function. April Fools always brings peak creativity.

#Riemann #Zeta #ZetaFunction #RiemanZetaFunction #AprilFool #AprilFools #AprilFoolsDay #Creativity #Math #Maths #NumberTheory #PeakCreativity #Nerd #Nerds #Humor #Humour #Alien #AlienSignals

While writing this Quora answer I wanted to try to implement the algorithm, or at least run a nontrivial example:
https://www.quora.com/Lagrange-showed-that-every-positive-integer-N-could-be-represented-by-the-sum-of-at-most-four-integer-squares-Is-there-an-efficient-algorithm-for-a-given-N-to-determine-the-values-of-these-four-or-fewer-components/answer/Mark-Gritter

The question asks for an algorithm to find four squares that sum to a given integer; their existence is given by Lagrange's four squares theorem. There's a very nice randomized-time algorithm in this paper: https://onlinelibrary.wiley.com/doi/10.1002/cpa.3160390713 (where a few more are discussed, whose status depends on unsolved number theory conjectures.) It uses the Hurwitz quaternions, which are the ring of quaternions with integer coefficients or entirely half-integer coefficients.

Given an odd n, we want to to find a pair of numbers x^2 + y^2 = -1 mod n, then compute gcd( x + yi + j, n ). If the gcd has norm n, it's a solution. Otherwise the norm divides n and gives us two recursive cases, which can be combined with quaternion multiplication.

The problem is finding an implementation of Hurwitz division to perform the GCD. No luck with Sympy and as usual I couldn't make heads or tails of the Sage documentation. I found a Python package that claimed to be an implementation of Hurwitz quaternions, but its division algorithm didn't seem to work-- an open bug says multiplication is wrong! https://github.com/HARP-research-Inc/hurwitz/issues/3

Finally, Wolfram came through; the GCD (both left and right) is part of this package: https://reference.wolfram.com/language/Quaternions/tutorial/Quaternions.html.en So I could work an example in a notebook.

I didn't bother looking for a pre-canned algorithm for the modular sum of squares problem -- brute force was good enough for a small example.

It'd be nice to revisit this and put the pieces together in a full implementation, but it was just too much work for the time I wanted to spend on it this evening. :)

#NumberTheory

The Peano Axioms: Building Blocks of Arithmetic

https://principlesofcryptography.com/number-theory-primer-an-axiomatic-study-of-natural-numbers-peano-axioms/

#HackerNews #PeanoAxioms #Arithmetic #NumberTheory #Mathematics #Logic

Happy silly 3.14159265358979323846 Friday.

P.S. if the number is wrong, blame chatgpt

#Friday #numbertheory #mathhumor

Online Codes & Expansions (CodEx) Seminar Tues Mar 11 2025

Steve Flammia presents
"A Constructive Approach to Zauner’s Conjecture via the Stark Conjectures" using #NumberTheory to build #quantum SIC-POVMs.

Based on https://arxiv.org/abs/2501.03970 joint with Marcus Appleby and Gene Kopp.

Register for the zoom link here: https://www.math.colostate.edu/~king/codex/

#seminar #online #math

Spectacular Connection Between #LLMs, #QuantumSystems, and #NumberTheory

DataScienceCentral.com https://www.datasciencecentral.com/spectacular-connection-between-llms-quantum-systems-and-number-theory/

#Mathematics #AI

Have you ever wondered what the first million digits of π↑↑4 might be?
Well, wonder no more...
https://ankokudan.org/d/d.htm?mathlistindex-e.html
#maths #constants #NumberTheory

The #Euclid-#Euler #Theorem is a fundamental theorem in #NumberTheory that beautifully connects #PerfectNumbers with #Mersenne #Primes.

https://knowledgezone.co.in/kbits/679340ac379a5d09cb2d1a93

Russian mathematician Ivan Mikheevich Pervushin, born OTD in 1827, discovered the 9th Mersenne prime, and proved that the 12th and 23rd Fermat numbers were composite https://cromwell-intl.com/travel/russia/getting-there.html?s=mb #travel #russia #NumberTheory

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

I've seen posts in various places about the properties of the number 2025, but none of them were as comprehensive as this.

* 45²
* 9² x 5²
* 40²+ 20²+5²
* 1³+2³+3³+4³+5³+6³+7³+8³+9³
* (1+2+3+4+5+6+7+8+9)²
* 1+3+5+...+89

#NumberTheory
#NumberTerrorist

For anyone interested, here are links to the relevant sections of Wikipedia pages for the proofs in order:

1. Well-Ordering Principle
https://en.wikipedia.org/wiki/Well-ordering_principle#Properties

2. Euclidean Division
https://en.wikipedia.org/wiki/Euclidean_division#Proof

3. Bézout's identity
https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity#Existence_proof

4. Euclid's Lemma
https://en.wikipedia.org/wiki/Euclid%27s_lemma#Using_B%C3%A9zout's_identity

5. Unique Prime Factorization
https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic#Proof

#math #proof #NumberTheory #prime

For a long time, I didn't know exactly how the Fundamental Theorem of Arithmetic (unique prime factorization) was proved from first principles.

I finally figured out the order of proofs for the fundamental theorem of arithmetic from first principles. Now that I know the sequence, it feels like it was my own fault for not properly going through a textbook and that it was probably obvious to everyone. I'm posting this mainly to reinforce my own clarity on the topic.

1. Well-Ordering Principle, this is either taken as axiomatic or one step away from axioms. This is where I'm drawing the line for "first principles".
2. Euclidean Division: Existence and Uniqueness of quotient 𝑞 and remainder 𝑟 in 𝑎 = 𝑞𝑏 + 𝑟 form. Uniqueness if 𝑟 between [0,1,...,𝑞−1]. Proof is just well-ordering principle (often framed as infinite descent).
3. Bezout's Identity. Proof starts with well-ordering to find 𝑑, then together with Euclidean division shows that 𝑑 is a divisor. A bit more Euclidean division shows that all other factors 𝑑' also divide 𝑑. So 𝑑 is the greatest divisor.
4. Euclid's Lemma: Proof says if 𝑝 doesn't 𝑎 in product 𝑎𝑏, then gcd(𝑎,𝑝)=1. This brings in the above and some rearrangement proves that 𝑏 is a factor as needed.
5. Fundamental Theorem of Arithmetic: Existence usually goes via strong induction (which can be rephrased as Well-Ordering + contradiction). Uniqueness uses Euclid's Lemma to show that the primes in the factorization must divide each other (which implies equality).

#math #proof #NumberTheory #prime

TIL: There is (in a sense) such a thing as a most divisible number

https://www.youtube.com/watch?v=V0IAtt5cBxY

#math #NumberTheory

"We could classify any area of math we think is leading in a bad direction to make it a state secret and "it will end"."

My decomposition is a state secret.
Academia is 13 years late.

#decompwlj #math #mathematics #maths #OEIS #sequences #numbers #primes #PrimeNumbers #FundamentalTheoremOfArithmetic #NumberTheory #graph #classification #integer #decomposition #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #academia #research #weight #level #jump

TIL: Fermi-Dirac-Primes.
(Not primes, but you can multiply them to get any integer. Construcing a number this way, any f-d-prime will occur at most once as factor. This has been compared to fermion behavior, and hence the name.)
https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_prime
2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, ...
#math #NumberTheory

A #geometry masterpiece: #Yale prof solves part of math’s #RosettaStone.

Yale’s #SamRaskin has solved a major portion of a math question that could lead to a translation #theory for some areas of #math.

❛❛ The [Robert] #Langlands Conjectures … suggested in the 1960s that deep, unproven connections exist between #numbertheory, harmonic analysis & #geometry — 3 areas of math long considered distinctly separate. ❜❜

https://news.yale.edu/2024/11/01/geometry-masterpiece-yale-prof-solves-part-maths-rosetta-stone 01 Nov 2024
https://Wikipedia.org/wiki/Langlands_program

#KrononautMoon

“Math Is Still Catching Up To The Mysterious Genius Of Srinivasa Ramanujan”, Quanta (https://www.quantamagazine.org/srinivasa-ramanujan-was-a-genius-math-is-still-catching-up-20241021/).

On HN: https://news.ycombinator.com/item?id=41909564

#Math #Mathematics #NumberTheory #Ramanujan #SrinivasaRamanujan #Genius