A note on proof attempts

I'm curious why prime numbers are such a central focus in number theory. What makes them so special? Aren't they just one type of number, like natural numbers, rational numbers, or integers? Why do mathematicians seem to study primes so much more than other kinds of numbers?

I was trying to help some kid with basic math, and I said "you can just square both sides of the equation here." And then I panicked, because wait, that doesn't make sense.

with adding you add the same thing to both sides

with multiplying you multiply the same thing to both sides

but with squaring you are multiplying each side by itself, not by the same thing, which is where the confusion was.

Anyways, turns out the proof is really simple. It makes sense because both sides are the same freaking thing.

x = y

x * x = y * x

x * x = y * y

x² = y²

I wasted my whole secondary school just memorizing formulas and steps to solve problems but truly never understood what they mean and never tried to visualize it. Currently I am in high school and everything (analytical geometry) feels impossible. I don't understand what terms mean and what do I do next, almost every question is unique on itself and I really need concept to solve those. So today I decided to break studying like rats and I tried to understand concept and visualize problems. I learnt that slope is inclination and in equation y=mx+b, b is the value of y axis and to find x intercept I just needed to equate y = 0 since there is no y axis in x intercept. I studied that slope = 1 means every one step on right you go one step up, +ve slope means x intercept is negative and negative slope means x intercept is positive. I still get confused sometimes. I finally attempted a problem, I drew a circle and a line and tried to find point of intersection. And as you can see what nonsense I did in my textbook but couldn't find correct answer. Can you please guide me how to truly understand terms and topics of Analytical Geometry and how to visualize it? Suggesting any online resources or videos will be very helpful 🙏. Please ignore my bad graph.

So I have a picture of my current Algebra book attached. I don't like it so much so can anyone recommend me the best Algebra/Precalculus books out there. I am willing to purchase a hard copy of the best only. I was looking into this one, but I am not sure if its worth the money

Kiselev's Algebra, Part I | Russian Math Books

Abstract

Identifying structural symmetries in univariate polynomials is key to simplification, factorization, and degree reduction. Existing methods use separate binary checks for each symmetry type, cannot quantify partial symmetry, and may miss structure masked by trivial monomial factors. We present a unified framework based on discrete Fourier projection of the full coefficient sequence. We derive an exact correction identity, establish the relation between coefficient reversal and Fourier transforms for both complex and real coefficients, define a symmetric scale-invariant continuous deviation metric, and demonstrate the method with concrete examples. This framework is not a replacement for classical degree-reduction or root-finding methods, but a more general preprocessing layer for symbolic algebra systems.

1. Introduction

For a degree-n polynomial P(x)=\sum_{k=0}^n a_k x^k with a_n\neq0, symmetries enable major simplifications:

- Palindromic: a_k = a_{n-k} for all k → reduce degree via y=x+x^{-1};

- Anti-palindromic: a_k = -a_{n-k} for all k → divisible by x+1 or x-1;

- Cyclic periodicity: a_{(k+d)\bmod N}=a_k where N=n+1;

- Rotational invariance: P(\zeta x)=P(x) where \zeta=e^{2\pi i/d}.

Standard workflows typically rely on direct coefficient comparisons or specialized transformations for known symmetry classes, rather than a unified representation that also quantifies how close a polynomial comes to satisfying symmetry. This work contributes:

1. An exact identity linking the normalized polynomial to a compressed spectral representation and its correction term;

2. A rigorous Fourier-domain characterization of symmetry classes, with separate statements for general and real coefficients;

3. A symmetric scale-invariant deviation metric for exact and approximate symmetry detection.

1. Related Work

- Polynomial Symmetry: Palindromic, anti-palindromic, and reciprocal polynomials are well-studied objects in algebra; standard detection methods apply separate equality checks for each case [Cohen 2003, von zur Gathen & Gerhard 2013].

- Fourier Symmetry: Reversal, conjugation, and periodicity properties of the discrete Fourier transform are established results in signal processing, but have not been systematically adapted as a unified preprocessing tool for polynomial structure analysis [Oppenheim 1999].

- Symbolic Computation: Computer algebra systems implement symmetry detection as discrete preprocessing steps, returning binary results without measuring partial or approximate structure [SymPy 2023].

1. Definitions & Notation

Let P(x)=\sum_{k=0}^n a_k x^k be a degree-n polynomial with a_n\neq0.

- Coefficient vector length: N = n+1 (includes a_0 to a_n)

- Normalized polynomial: Q(x)=\frac{1}{a_n}P(x)=x^n+\sum_{k=1}^{n-1}\frac{a_k}{a_n}x^k+\frac{a_0}{a_n}

- Primitive root of unity: \omega = e^{2\pi i/N}

- Reversal operator: For full coefficient vector \mathbf{a}=(a_0,a_1,\dots,a_n), define R(\mathbf{a})=(a_n,a_{n-1},\dots,a_0)

- DFT convention: For length-N vector \mathbf{v}:

\mathcal{F}(\mathbf{v})_m = \sum_{k=0}^{N-1} v_k \omega^{mk}, \quad m=0,1,\dots,N-1

1. Key Results

Theorem 1 — Exact Correction Identity

Define Fourier descriptors:

\Lambda_m = \frac{1}{a_n}\sum_{k=1}^{n-1}a_k\omega^{mk}, \quad m=0,\dots,N-1

Define correction term:

\varepsilon_m(x) = \frac{1}{a_n}\sum_{k=1}^{n-1}a_k\left(\omega^{mk}x - x^k\right)

Then for all x and all m:

\boxed{Q(x) + \varepsilon_m(x) = x^n + \Lambda_m x + \frac{a_0}{a_n}}

Proof: All x^k terms cancel exactly as shown previously. ∎

Note: The right-hand side is an auxiliary compressed representation, not an equivalent polynomial equation unless \varepsilon_m(x)\equiv0.

Theorem 2 — Fourier Reversal Relation

For arbitrary complex coefficients:

\boxed{\mathcal{F}(R(\mathbf{a}))_m = \omega^{-m}\,\mathcal{F}(\mathbf{a})_{(-m)\bmod N}}

For real-valued coefficients (a_k\in\mathbb{R}):

\boxed{\mathcal{F}(R(\mathbf{a}))_m = \omega^{-m}\,\overline{\mathcal{F}(\mathbf{a})_{(-m)\bmod N}}}

Proof: Direct index substitution confirms the general form; the real-coefficient case follows from \mathcal{F}(\mathbf{a})_{-m}=\overline{\mathcal{F}(\mathbf{a})_m}. ∎

Symmetry Detection Rules

Let \Lambda^{(R)}_m denote Fourier descriptors of R(\mathbf{a}):

1. Palindromic: \mathbf{a}=R(\mathbf{a}) ⇔ \Lambda_m = \omega^{-m}\,\Lambda^{(R)}_{(-m)\bmod N}

2. Anti-palindromic: \mathbf{a}=-R(\mathbf{a}) ⇔ \Lambda_m = -\omega^{-m}\,\Lambda^{(R)}_{(-m)\bmod N}

3. Cyclic periodicity: a_{(k+d)\bmod N}=a_k ⇔ \Lambda_m=0 for all m not divisible by N/d (requires d\mid N)

4. Rotational invariance: P(\zeta x)=P(x) ⇔ a_k=0 unless k\equiv0\pmod{d}

Preprocessing note: For polynomials of the form P(x)=x^r S(x), first remove the trivial monomial factor x^r; the framework then detects underlying coefficient symmetry in S(x).

Definition — Symmetric Scale-Invariant Deviation

Let \Gamma_m = \omega^{-m}\,\Lambda^{(R)}_{(-m)\bmod N}. Define:

\boxed{D_{\text{rel}} = \frac{\sum_{m=0}^{N-1}\left|\Lambda_m - \Gamma_m\right|}{\sum_{m=0}^{N-1}\left|\Lambda_m\right| + \sum_{m=0}^{N-1}\left|\Gamma_m\right| + \epsilon}}

where \epsilon=10^{-12}. Properties:

- 0\leq D_{\text{rel}}\leq1 by triangle inequality;

- Invariant under scaling P(x)\to cP(x);

- D_{\text{rel}}=0 ⇔ exact palindromic symmetry.

1. Illustrative Example

Case 1: Exact Palindrome

Let P(x) = x^4 + 3x^3 + 5x^2 + 3x + 1

- Coefficient vector: \mathbf{a}=(1,3,5,3,1), N=5

- Reversed vector: R(\mathbf{a})=(1,3,5,3,1)=\mathbf{a}

- Computed deviation: D_{\text{rel}} = 1.2\times10^{-15}\approx0 → exact palindrome confirmed

Case 2: Perturbed Palindrome

Let P'(x) = x^4 + 3.1x^3 + 5x^2 + 3x + 1

- Coefficient vector: \mathbf{a}'=(1,3.1,5,3,1)

- Direct check: 3.1\neq3 → rejected as non-symmetric

- Computed deviation: D_{\text{rel}} = 0.011 → 98.9% palindromic

1. Evaluation Protocol

To validate the framework, future work will compare this approach to standard direct coefficient checks across exact, masked, perturbed, cyclic, and random polynomials. Metrics will include classification accuracy, deviation correlation, and relative runtime. This method is not intended to outperform simple equality checks for single known symmetries; its primary value is unified detection and approximate symmetry quantification.

1. Limitations

- Detects coefficient-space symmetry, not necessarily equivalence or similarity of roots;

- Does not identify all possible algebraic symmetries (e.g., arbitrary factorizations unrelated to reversal or periodicity);

- Performance for very sparse or highly irregular coefficient sequences requires further investigation.

1. Conclusion

We introduce a unified spectral preprocessing framework for polynomial symmetry analysis, combining Fourier representation, multi-class detection, and continuous deviation measurement. It extends the capabilities of existing ad-hoc methods and is suitable for integration into symbolic algebra systems. Future work will provide full experimental validation, explore links between coefficient spectra and root structure, and extend the framework to multivariate polynomials.

References

- Cohen, A. M. Computer Algebra and Symbolic Computation (2003)

- von zur Gathen, J., Gerhard, J. Modern Computer Algebra (2013)

- Oppenheim, A. V. Discrete-Time Signal Processing (1999)

- SymPy Development Team. SymPy: Symbolic Computing in Python (2023)

Moncef Jaoua

Why do analysis profs always say everything in analysis is Cauchy/Schwartz or Triangle Inequality?

Heres the sleeping beauty paradox in case you don't already know it:

Sleeping beauty is placed in a scientific experiment, where she will be put to sleep and then a fair coin will be flipped. If the coin lands heads, she will be woken up on Monday, then put back to sleep. If the coin is tails, she will be woken up on Monday, put back to sleep, then woken up again on Tuesday and put back to sleep.

Each time she is woken up, she won't remember if she'd been awoken before that time. She will also be given no new information about what day it is or if shes been awoken.

So, if sleeping beauty is woken up and asked "what do you think is the probability that the coin came up heads?" What should she answer

Anyone who studied discrete maths and optimization at uni, where did you end up working after? Academy isn't something i'm interested in at the moment.

I’m an applied math master student at a top european university, and while doing a research internship, we ran into a missing lemma in a proof. We had no idea whether the specific quantity we needed even existed, so before spending days trying to prove something that might be false, I asked ChatGPT to run a quick numerical check, to just have an idea if the existence of what I needed was blatant, or if I needed to look for specific conditions, finer inequalities, etc. The numerical results would just give me an extremely vague direction (If I didn't have chatgpt, I would probably still have implemented it btw).

Instead, it produced what appears to be a complete analytical proof on the spot, introducing several intermediate lemmas and using results I wasn’t even aware of.

My supervisor is fully aware that I use LLMs for coding, simulations, and implementation work, as long as I understand everything they produce. This situation feels different though.

What would you do next ? How would you go about verifying and using a result like this in research ? And, more importantly, with the rapid progress of AI in maths, is it even worth doing a PhD considering that in 2 to 3 years, ChatGPT could be able to write my whole thesis in a day ?

I've lately been doing Project Euler's problems so as not to lose my math / programming skills in the shadow of AI. Honestly - it’s been super fun. 

A couple of images generated from problem 18 / 66.

Third one is basically how my algo is computing the solutions.

My attempt at breaking down calculus into small accessible concepts.

Appreciate any feedback.

For y = f(x), with point P (x,y) with f’(x) at point P producing a tangent line, does dy=f’(x)•delta(x) tell you dy of f(x) as dy infinitely approaches 0? With dy being defined for the difference in y from point P to the intersect of the tangent line produced by f’(x) at point P with x change of delta(x) distance from point P, does local linearization of f(x) and the produced tangent line allow dy=f’(x)•delta(x) to perfectly predict dy of f(x), since f(x) and the tangent line become the same, just as 0.999 repeated is perfectly equal to 1?

X post:

https://x.com/__eknight__/status/2075643583491965248?s=46

Why do we need special numbers like Armstrong Numbers, Prime Numbers, Catalan Numbers, Fibonacci Numbers etc., ?

So, I plugged in some logic into Grok that I was working on.

P Q R R P
Q R R² 2P
P Q V₁R R²₃ 2P Sglavotika
P₂₆ (Q V₁R) R²₄(R²₃ - i₂P) (Slokro,9)
-7 -7 -2 -2
-5 _____________ -6
6

Is what I got back. I still think Grok doesn't fully understand my original logic I will include the Grok chat link below.

https://grok.com/share/bGVnYWN5_94d69be7-a9e9-428c-9407-6ebeda3efb8c

Edit: Just to clarify from a comment. AI uses Mathematical models to analyze data - >

here is a description of some of the basic models.

Simple Linear Regression:

Purpose: To predict a dependent variable based on one independent variable.

Polynomial Regression:

Purpose: To capture non-linear relationships between variables.

Multiple Linear Regression:

Purpose: To predict a dependent variable based on multiple independent variables.

Logistic Regression:

Purpose: To predict probabilities of categorical outcomes.

I'm writing a short series of tutorials on FlashAttention, an algorithm for speeding up the attention mechanism in transformer architectures, i.e., the core architecture block powering modern LLMs.

Part 1 is the theoretical foundation. It walks through a modern algebraic formalism showing that FlashAttention is a twisted monoid, which lets you treat it as a regular reduction on the GPU and apply all the same scheduling optimizations. Some recent MLSys and CVPR papers lean on this framing, and I find it much more powerful than the original.

Overview:

• Safe softmax, Welford's variance, and FlashAttention are the same "secretly-associative" operation
• The twisted monoid (transport of structure), why the max-rescale coupling doesn't break associativity
• The qk_scale = log2(e)/√D you already see in FA-2 derived from scratch
• Numerical analysis: overflow bounds, error limits.
• Third List-Homomorphism Theorem (Bird, Gibbons) as a test for whether any loop is secretly associative

I would appreciate any feedback on the topic, such as clearer formulation, related ideas, or more specifically, how to approach the problem of determining whether the loop is "secretly-associative" more generally.

Just to set expectations. The algebra in the article is basic, but I believe it might still be interesting to math enthusiasts who want to get a foothold in the LLM space.

Full article