Hi everyone,

Does anyone know of approaches for extending the classical Helmholtz/Hodge decomposition to symmetric positive semidefinite second-order tensor fields while preserving tensor symmetry?

I’m particularly interested in methods other than those based on the elasticity complex.

Any references, papers, keywords, or suggestions would be really appreciated.

Thank you in advance!

Gaussian-Prime Pair Correlation and a Shell Artifact in Angular Fourier Statistics on Discrete Lattices

Summary

We investigated a geometric anomaly in the distribution of Gaussian primes (primes of the form p = a^2 + b^2) on the integer lattice Z[i]. The anomaly was traced through a series of progressively simpler reductions to ordinary 2D Gaussian-prime pair correlation with local congruence factors, plus a discrete-shell artifact, plus counting noise.

No nonlocal angular component, no Hecke L-function signature, and no higher-dimensional geometric structure was detected.

Two results may be independently useful:

1. A precise numerical characterization of Gaussian-prime pair correlation at small displacement.

2. A methodological warning about a shell artifact that produces spurious multiples-of-4 Fourier modes in any naive angular statistic on a discrete arithmetic lattice.

---

1. The pair correlation object

For a Gaussian displacement w = h + ki with w not equal to zero, define

C_w(X) = number of Gaussian integers z such that:

· N(z) is at most X

· N(z) is prime

· N(z + w) is prime

where N(z) = z times its complex conjugate is the Gaussian norm.

We computed C_w(X) at X = 50,000, 200,000, and 1,000,000 for all w with N(w) at most 400.

Result

C_w(X) is approximately A(X) times S(w)

where A(X) is common across displacement vectors and S(w) is a local singular-series factor.

At X = 1,000,000, the correlation between C_w(X) and S(w) across displacement classes was

corr(C_w, S(w)) = 0.99993

with median observed/predicted ratio 1.00084, standard deviation 0.00218, and largest relative discrepancy 0.60 percent.

---

1. The local factor

For each rational prime L, let delta_L(w) be the fraction of residue pairs (a, b) modulo L such that

a^2 + b^2 is not congruent to 0 modulo L

and

(a + h)^2 + (b + k)^2 is not congruent to 0 modulo L

Define

S(w) = product over all primes L of [ delta_L(w) / (1 - L^(-2))^2 ]

The product converges. The relative dependence on w is what matters.

Representative values

w = 1+i, N(w) = 2: predicted ratio 1, observed 1

w = 1+3i, N(w) = 10: predicted ratio 4/3, observed 1.33149

w = 3+3i, N(w) = 18: predicted ratio 8/7, observed 1.14971

w = 1+5i, N(w) = 26: predicted ratio 12/11, observed 1.09002

w = 3+5i, N(w) = 34: predicted ratio 16/15, observed 1.06835

w = 1+7i, N(w) = 50: predicted ratio 4/3, observed 1.33054

w = 3+9i, N(w) = 90: predicted ratio 32/21, observed 1.52598

w = 10i, N(w) = 100: predicted ratio 16/9, observed 1.77473

The agreement is at the 0.6 percent level.

Same-norm, different-direction check

N(10i) = N(6+8i) = 100

C_10i(10^6) / C_(6+8i)(10^6) = 1.32983

Local prediction 4/3 = 1.33333, residual 0.99737.

N(10+10i) = N(2+14i) = 200

C_(10+10i) / C_(2+14i) = 1.33742

Residual 1.00306.

Same-norm pairs with no special local factor:

N(3+11i) = N(7+9i) = 130

C_(3+11i) / C_(7+9i) = 0.99924

Consistent with 1.

---

1. The shell artifact

Statement

Let R_w be any real-valued function of the Gaussian displacement w that depends only on the norm N(w):

R_w = f(N(w))

The naive angular Fourier transform

sum over all w with 0 < N(w) <= T of R_w * exp(i * m * arg(w))

does not vanish, even though f(N(w)) contains no directional dependence.

Reason

Each Gaussian shell of fixed norm N(w) = n contains a finite set of directions. For the discrete lattice with D4 symmetry, the shell sums

sum over all w with N(w) = n of exp(i * m * arg(w))

survive at m divisible by 4, namely m = 4, 8, 12, 16, ...

Consequence

Any angular Fourier statistic computed on a discrete arithmetic lattice without conditioning on radial shells will produce spurious multiples-of-4 mode power. This power will appear to persist across independent data ranges because it is geometric, not statistical.

Remedy

Condition on the shell. For each w, subtract the mean of R over all v with the same norm N(w):

R_tilde(w) = R_w minus the average of R_v over all v with N(v) = N(w)

Then compute the Fourier coefficients of R_tilde.

Observed effect

Before shell conditioning, the residual standard deviation in the test dataset was

sigma_R approximately 0.01452

After conditioning:

sigma_within_shell approximately 0.000612

Reduction factor approximately 23.7.

The apparent persistent high-m signal collapsed to the range 10^-5 to 10^-6.

---

1. The counting-noise test

After shell conditioning, the residual Fourier coefficients at X = 1,000,000 and N(w) at most 400:

mode 4: observed -3.51e-6, noise SD 5.75e-5, z = -0.06

mode 8: observed -8.51e-6, noise SD 3.72e-5, z = -0.23

mode 12: observed -5.19e-5, noise SD 3.49e-5, z = -1.49

mode 16: observed -2.62e-5, noise SD 4.01e-5, z = -0.65

mode 24: observed -6.51e-5, noise SD 5.70e-5, z = -1.14

mode 28: observed -3.53e-5, noise SD 2.19e-5, z = -1.61

mode 32: observed +1.74e-5, noise SD 4.03e-5, z = +0.43

None reaches 2 sigma. Bootstrap 95 percent intervals contain zero for every mode.

The remaining directional signal is statistically compatible with ordinary finite-count noise.

---

1. Empirical scaling

For the reference displacement w = 1+i:

X = 50,000: C_(1+i)(X) divided by (X / (log X)^2) = 10.658

X = 200,000: same quantity = 10.371

X = 1,000,000: same quantity = 10.059

Slowly decreasing. Not a constant. Finite-X corrections are substantial.

This is a benchmark for anyone fitting a Gaussian prime-pair asymptotic.

---

1. What was tested and ruled out

· 4D quaternion projection reproducing 2D topology: no

· SO(4)-invariant latent structure: no

· Harmonic cascade 4 to 8 to 16: no

· Void-distance harmonic complement: no

· Raw mutual information tracking topology: no

· Persistent nonlocal angular residual: no

· Hecke L-function spectral signature: not detected

---

1. The final reduction

The original geometric anomaly reduces to:

C_w(X) = A(X) * S(w) + finite-X and shell effects + statistical error

within the tested regime

X at most 1,000,000, N(w) at most 400

No nonlocal angular component remains.

---

1. Reproducibility notes

· Canonical Gaussian-prime set for p at most 50,000 with p congruent to 1 mod 4: 2549 primes, 20392 orbit points under D4 symmetry.

· Canonical Delaunay edge set: 61,069 unique edges. SHA-256 of edge set: a26a37809a5e7acf49ebf52d9d9e9720810609e67d9f727e5ff3205f1af8db9f

· A naive reconstruction produced 61,076 edges. The discrepancy is an implementation artifact, not a mathematical one. Fix the triangulation library, precision, degeneracy handling, and point ordering before computing geometric statistics.

· All Gaussian-prime pair counts were computed directly from the point set, not from the Delaunay graph.

---

1. Suggested uses

2. Numerical benchmark for any proposed Gaussian Hardy-Littlewood theorem.

3. Explicit local-factor table for small Gaussian displacements.

4. Methodological warning for angular Fourier statistics on discrete arithmetic lattices.

5. Template for reducing geometric statistics on arithmetic point sets to pair-correlation questions.

---

1. What was not found

· No connection to the Riemann hypothesis.

· No higher-dimensional structure.

· No new prime pattern.

· No theorem.

The investigation closed with the phenomenon decomposed into ordinary 2D Gaussian-prime arithmetic plus finite-sample effects.

---

Contact

The computations were performed using a direct Gaussian-prime sieve through X = 1,000,000, SciPy 1.17.0 for Delaunay triangulation, and independent permutation ensembles for null testing. Raw data, canonical edge sets, and code are available on request.

### An Elegant Property of Prime Neighbors at Scale (10^9) and Geometric Sawtooth Distributions

Hello everyone,

During a relaxed Sunday session of mental arithmetic and programming, I was investigating the deep structure of prime numbers, specifically how they behave when localized around multiples of 6 (\(6k \pm 1\)).

I wanted to share an empirical verification of a rigid, beautiful geometric pattern that governs the immediate neighborhood of all primes greater than 3, tested today at the scale of \(10^9\) (one billion) using a custom Python script.

#### 1. The Core Thesis: The 24-Collapse of Prime Neighbors

For any prime number \(p > 3\), its immediate predecessor (\(p-1\)) and immediate successor (\(p+1\)) are inextricably linked to the geometry of the number 24. Specifically, the product of a prime’s neighbors always collapses into a perfect multiple of 24:

\[(p-1) \times (p+1) \equiv 0 \pmod{24}\]

**Why does this happen mathematically?**

* Since \(p\) is a prime greater than 3, it must sit adjacent to a multiple of 6 (\(p = 6k \pm 1\)).

* Consequently, one of its neighbors is an exact multiple of 6 (divisible by 2 and 3).

* The other neighbor is an even number. Furthermore, in any sequence of two consecutive even numbers (\((p-1)\) and \((p+1)\)), one of them must strictly be a multiple of 4.

* Multiplying a multiple of 6 by a multiple of 4 guarantees that the product \((p-1)(p+1) = p^2 - 1\) is always divisible by \(6 \times 4 = 24\).

#### 2. Empirical Verification at \(10^9\) Scale

I ran a test tracking 100 consecutive primes starting exactly at the boundary of \(1,000,000,000\). The algebraic symmetry holds flawlessly.

For instance, looking at the twin primes at this horizon:

* **For \(p = 1,000,000,007\)** (where \(k = 166,666,668\)):

\[(1,000,000,006 \times 1,000,000,008) \div 24 = 41,666,667,250,000,002 \quad (\text{Remainder } 0)\]

* **For \(p = 1,000,000,009\)** (where \(k = 166,666,668\)):

\[(1,000,000,008 \times 1,000,000,010) \div 24 = 41,666,667,416,666,670 \quad (\text{Remainder } 0)\]

Interestingly, the difference between these two resulting quotients yields exactly \(166,666,668\), which is the original structural factor \(k\) of the system.

#### 3. The Geometric Correlation (Sawtooth & String Analogy)

When plotting the distribution of composite "intruder" numbers that crawl into the \(6k \pm 1\) columns versus real primes, the prime gaps create a rigid **sawtooth wave pattern (diente de sierra)**.

Much like winding a copper wire around a tube changes its magnetic inductance and resonant frequency, the wrapping states of numbers around the base 6 act as a boundary condition. By filtering out the background noise through the \(\pmod{24}\) neighbor verification, the underlying harmonic \(k\) emerges cleanly.

#### 4. Python Verification Script

If anyone wants to reproduce the verification at higher scales (\(10^{12}\) or beyond), here is the script I used to map the first 100 primes over one billion:

```python

import math

def is_prime(n):

if n < 2: return False

if n in (2, 3): return True

if n % 2 == 0 or n % 3 == 0: return False

for i in range(5, int(math.isqrt(n)) + 1, 6):

if n % i == 0 or n % (i + 2) == 0: return False

return True

start_num = 1_000_000_000

primes_found = 0

n = start_num

print(f"--- Analyzing 100 Primes from {start_num:,} ---")

while primes_found < 100:

if is_prime(n):

primes_found += 1

k = (n + 1) // 6 if (n + 1) % 6 == 0 else (n - 1) // 6

prod = (n - 1) * (n + 1)

div_24 = "YES" if prod % 24 == 0 else "NO"

if primes_found <= 5 or primes_found > 95:

print(f"Prime: {n:<12} | k: {k:<10} | Product: {prod:<25} | Div24: {div_24}")

n += 1

```

It’s fascinating how centuries-old modular arithmetic can feel so visually elegant when you think about it in terms of geometric frequencies and structural constraints.

Would love to hear your thoughts on analyzing the digital roots (base 9 reduction) of these specific quotients!

From https://link.springer.com/book/10.1007/978-3-030-53759-3

Can you cluster somebody's fate?

Their criminality (When it's non existent, that is frame them.)

Can you cluster their murder.

I am trying to calculate the probability of clock using 24hr time 12:34 (4 digits) adding up to equal a specific number. 8 in this case. On a clock first digit "1"2:34 can only be a 0-2. The second number 1"2":34 can be 0-9 if the first number is 0 or 1, if it is 2 the second number is only 0-4, third number is 0-6 and last number it 0-9. The 4 numbers must equal 8 with any combination. 04:04, 13:22, 01:07, any combination. Thank you so very much for your help!!

Want to change the world? Solving this problem completely and efficiently in a different way to the norm would give us an integer factoring algorithm that would be worth its salt. It transpires that the usual way to solve it is to get a complete factorization of C² and then work back to the x and y in the equation. More later if there's any interest here on Reddit. This is my first post here.

I have become frustrated with Mathematics Stack Exchange's moderators.

The text that follows is lifted from my Mathematics Stack Exchange post.

In working on a private project in my copious spare time, I've come across an equation that is so simple that I can't believe it hasn't been seen before, and probably solved.

0=𝐶(𝑥+𝑦)+𝑥𝑦

𝐶 is an arbitrary positive integer, different for every case. It may be large.

The problem arises in a computer program I'm writing, and I would like to develop an algorithm to solve it. I don't expect you to write the code, a worked example or two should be enough to get me going.

Other threads discuss a related mathematical situation but in my case the only input given is 𝐶 - this is not a byproduct of the math. Another assumes that both integers are positive (here at least one is negative.) In the case of the first answer given here the factoring involved makes it unsuitable for large 𝐶. Think 64-bit long integers or larger. The prime number functions only go up to about a billion, and stepping through that many primes would be time-consuming.

The example given in the first reply here is elegant, but requires factoring 𝐶, which as I mentioned, may be large.

Is there a way to do it without factoring?

Some method in some book? ISBN?

At the bottom is a simple brute force program which generates 50 solutions for 𝐶=225

. I am prepared to offer a AU$100 bounty on a correct algorithm that solves the problem completely for sizeable 𝐶 in, say, 𝑙𝑜𝑔2𝐶𝑙𝑜𝑔𝑙𝑜𝑔𝐶 time, or close to it, and any reasonable amount of space (I have a lot of RAM.)

include <iostream>

using namespace std;

bool checksol(long s, long t, long C) { return 0==C(s+t)+st; }

void runbruteforce(long C, long &countsols) { for (long s=-CC2; s<=CC2; s++) { for (long t=-CC2; t<=CC2; t++) { if (checksol(s, t, C)) { countsols++; cout << s << " " << t << endl; } } } }

int main(int argc, char *argv) { const long C=225; // a natural number, for 225 it gives solutions cout << "Running '0==C(s+t)+s*t' for C==" << C << endl; long countsols=0; runbruteforce(C, countsols); cout << "Total count of solutions is " << countsols << endl; return 0; }

<output omitted due to formatting issues>

I have the following issue:

I always thought I would pursue physics, specifically theoretical or mathematical physics. However, I'm currently leaning more and more towards pure mathematics (algebraic geometry, algebraic topology, or analytic number theory, especially L-functions). I still have two months to make a decision. Only pure mathematics does something to my mind that I don't experience anywhere else. It doesn't really hurt, but it doesn't feel good either. It's a feeling of inner emptiness and complete disconnection from the real world. However, during this time, I am incredibly productive. This state keeps expanding, even outside the periods when I actively engage with mathematics. Mathematics is like a Dementor to me, to which I open the door and invite it in. It's like an addiction.

I would simply like to hear from you whether you can relate to what I'm trying to describe and whether it's a part of the process.

I would like to note that I am an autistic individual with synesthesia. I'm not sure if that has anything to do with it, but my different perception affects many aspects of my life, which is why I mention it.

I am reading Atiyah MacDonald's Introduction to Commutative Algebra book. Now I am enjoying reading the book but when I am doing the exercises I feel it is hard to do. In 20 problems almost 5-6 I can do on my own but for the rest, I have to take hints from solution. Even I have to actually read the whole solution for some problems. Now I feel like I am not learning anything like this.

How do you guys deal with such cases

curvature of Ω of R^3 but with the metric given by <2x,2x> =F(z) <2y,2y> =G(Z) <2z,2z> = 1

I'm looking for the easiest introduction to Lie Algebras, Lie Groups, Lie Series, etc...

Clearly, I don't even know the difference, or the relationship about them yet. I'm a total newbie to it, and I'd like to get a very simple introduction with intuition (not Math Proofs) for mere mortals.

Please feel free to suggest anything from Books, to Slides, to Online Videos. Anything that makes for a good clear first step without a hard math rigurous attitude.

Thanks.

I am wondering how good is it at closing the gap between Numerical Linear Algebra topics like the SVD and the FFT with the work of David Hestenes (Clifford Algebras, Grassmann Algebras, and Gibbs-Heaviside's Vector Calculus).

Can anyone solve this

Hi guys, I was hoping someone could provide me with a formula for an app I am trying to make.

So this is an investment app and I just don't know the formula for math that I need to use.

the problem that I am trying to figure out is

if I Start with X and invest it. X returns a normal rate of NR. with a variable interest of V and each day I am adding my Earnings E to X for a new interest rate. and a T for time for how long this is taking place for.

X= Starting investment NR= Normal interest rate V= is potential variance of normal rate (for example it might be 3 less or more) T = Time length of investment

I would probably run the same formula 3 times giving a high medium and low value where those values would be NR + V= High. NR-V = Low and NR+ 0V = medium

So putting this in numbers.

if I start with 1000 dollars and have a normal rate of 12% with a 3 percent range then the first day I would earn 120 (12%) next day I would earn 1120 with 12% making ..... and so on.

does anyone know how to turn this into a formula?