Modularity theorem Puns

Link: https://tomrocksmaths.com/2020/09/24/fermats-little-theorem/

This is the fourth (and final) article in the 'Introduction to Group Theory' series written by Tom Rocks Maths intern Aditya Ghosh. The concepts of modular arithmetic, symmetry and the group axioms introduced in the earlier articles are used to construct a proof of Fermat's Little Theorem.

Article 1: Modular Arithmetic and Calculating Expenses https://tomrocksmaths.com/2020/08/25/using-modular-arithmetic-to-calculate-expenses/

Article 2: The Importance of being Symmetric https://tomrocksmaths.com/2020/09/03/symmetry/

Article 3: When do equations have solutions? (an introduction to Group Theory) https://tomrocksmaths.com/2020/09/10/when-do-equations-have-solutions-an-introduction-to-group-theory/

Aditya is a second year undergraduate student studying Maths at the University of Oxford.

Tom Rocks Maths is an outreach programme run by Dr Tom Crawford - an Early-Career Teaching and Outreach Fellow at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-crawford

[first] [prev] [next]

Ru'udamo'o was a former Executor covert action specialist. He was a shadow in the system, the suggestion of a shadow, a hint of an existence and he was fine with that. He had served the Unified Civilized Council's peoples for three thousand years. When the C3 and it's collateral damage started he had sought out the forbidden to consult. The Herd Stallion. The Herd Matron. He did not fear their power, he sought their wisdom to understand where they intended on guiding the Great Herd.

He, more than anyone, understood that the Civilized Council was nothing more than a brackish swamp slowly drying away to salt plains. That in a hundred million years his people had managed to do little more than destroy star systems to feed an overwhelming appetite they didn't even understand.

He had watched the C3 War with interest. For the first time in the Council's history they weren't facing off against a species that had barely developed space travel and stellar class combat weapons with only a few systems for resources and manufacturing. Every scrap of information he could gather on the Confederacy pointed to thousands of systems, multiple species, heavy industrial capacity and capability, massive shipyards, research programs, and the infrastructure to prosecute interstellar wars.

While his peers and superiors (or those who saw themselves as his superior) just hand-waved away the eight thousand years of interstellar travel for the lemurs, Ru'udamo'o examined it closely.

It was doing a deep data dive on the Confederacy that he discovered a simple fact that the lemurs and their allies ignored.

The longest period of peace the Confederacy had enjoyed was an astounding 63 days.

The Confederacy was made up of aligned systems, not a single system under the rulership of the Senate.

Ru'udamo'o had found out that there were nearly 58 human stellar organizations withing the Confederacy. The Mantids had three. The Treana'ad only one. The Rigellian Suarian Compact covered over a dozen species, many with their own stellar governments owning multiple stellar systems.

Even during the C3 War, the Confederacy was actively fighting in their own territory against 'Rogue Nation States"

I have lately been interested in the notion of quantum information, and how it seemingly has become an important cornerstone of modern physics, with the Black Hole paradox, the Bell inequalities etc.

Eventually I found out about the no-hiding theorem, which sounds like a conservation law to me. So, as mentioned in the title, is there a corresponding symmetry to this conservation law, according to Noether's theorem?

Hi, I'm trying to figure out how these values were calculated for a question on the Chinese remainder theorem: http://imgur.com/a/6DQZq

I've tried using Euclid's Algorithm (and going backwards), and also following youtube videos but they don't seem to be working? It always seems like the n in [m (mod n)] is always larger than the m which isn't the case in my question so I'm even more confused. Any help would be much appreciated! Thanks

I am a high school student who just watched a documentary on youtube about fermat's last theorem. In the film they talked about modular forms. Can someone explain them so I may understand and please explain how it along with elliptic curves were used in proving fermat's last theorem so I can understand. Math is a favourite of mine and I'd love to understand what it all is. Thank you! Edit: I currently finished algebra 2 and am moving on to precalculus. Yet I am self studying calc and have a good working knowledge of it. I understand complex numbers.

I am having trouble understanding modular forms and elliptic curves, and was wondering if someone could explain them in layman's terms; maybe relate them to the world somehow.

Schooling: I have gone through Algebra 2 and will be going into precalculus. I self studied calc though.

Thank You!

Things that an Engineer or Science Officer on a star ship might say.

1. We’ve got a Warp Core breach
2. Get the ion matrix modulator back online
3. The gravimetric stabilizer is loose
4. A quantum matrix relay
5. Reroute power to the forward flux capacitor
6. Reverse polarity to theta radiation
7. Demodulate the deflector emitters
8. The Tridium particles are unstable
9. A Resonance Distortion in the Inductors
10. The Plasma Manifold can’t handle the flux
11. We must calculate the iridium three-factor conductors
12. Converge the radiation filters to infra-segmentation
13. Ion-pressure has reached critical matrix
14. Re-integrate the strontium-capacitor to redux-level
15. Restart the master chronium disruptor
16. Maximum astro turbilator integrated
17. Quantum-seed charged to energetic maximum
18. Thorium-flux source has been dispatched
19. Thermal integration detected
20. Astral velocity converted
21. Detonate the Illudium Q-35 explosive space modulator
22. Reverse the polarity of the neutron flow (Dr. Who)
23. There’s a malfunction in the retro-encabulator
24. Circadian extrapolation of the interpolation theorem
25. Spatial transposition phenomena
26. Inner realm qualia
27. Tripolar amalgamation theory
28. Single equation theory of the universal containment field
29. Modular neural interface component
30. Bio-cellular granularity
31. Holodistortion dome
32. Ecolyptic nervescaping processes
33. Micro-ontological effect syndrome
34. Brainlock ameliorator device
35. Mindquake predictor array
36. Achronic sterilization dilator
37. Noctilucent telescopy
38. Thanoscopic analyzer
39. Biotech construction units
40. Temporal augmentation
41. Subneural traweling graft
42. Bioware contamination particles
43. Nanogenetic augmentation cords
44. Helical supersemetric ray-bands
45. Monitored Cryonic Mini-Kelvin Rating
46. Quadra-Capacitor Circuitry
47. Virtual locutor telepresence
48. Genotypic avatar
49. Subspeciation matrix grid
50. Neuro-linked psi-bands
51. Dillinger-drive transfield component
52. Deploying frictionless vortals now, Sir
53. Enhance the divergence coefficient
54. We need to integrate the molecular cospology
55. Reconscede the Heisenberg drive
56. Activate the Omega 13 device
57. Reconfigure the nanowave frequency shift
58. You need to calibrate your ventral defragmenter
59. Reroute the secondary vacuum node
60. We’ve got a destabilized gluon pulse
61. Did you fractionalize the redundancy emitters?
62. I’m detecting a particle trace in the thr

So my math book does a terrible job at explaining how Euler's Theorem can be used to speed up modular arithmetic with large exponents.

I need to figure out the following using Euler's Theorem:

(12^320) mod 20

(5^210) mod 12

Solution and explanation would be GREATLY appreciated!

If I were to overly simplify it, basically if you assume the equation to be true, you can prove that it forms a certain type of mathmatical curve which has properties which make it impossible?

Hej out there. I have a assignment, and two of the questions are bugging me. 1. show that in the equation x^n + y^n =z^n have a (none trivial) whole number solution, so x^n + y^n =z^n ( mod p) for a prim number p, but not the other way around 2. show that Fermats last theorem, can be shown by x^4 + y^4 =z^4 and x^p + y^p =z^p (mod p), where p is a odd prim. I really need some help. Thanks. from a lost senior year highschool student.

So I want to be able to compute something crazy like 36^36 mod 11 using Fermat's theorem (b^p-1 mod p = 1, as long as p is prime and b,p are relatively prime). So for this problem, that would mean 36^10 mod 11 = 1. So I guess it's all about rearranging 36^36 mod 11 until I can get it into the form of 36^10 but I dunno how to do that or what to do afterward. Can someone explain this to me? Many thanks, reddit.

I know there is some overlap between both areas (eg L-functions, Dedekind zeta functions and the class number formula). But is there any reasearch on how algebraic NT tools can be used to solve analytic NT problems or viceversa? Or maybe some research area that uses both analytic and algebraic methods?

I am asking because I am about to apply for PhDs, and my interest is both algebraic and analytic NT. So I would love to get to know more about possible research directions.

Do your worst!

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

It really does, I swear!

They’re on standbi

Pilot on me!!

Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.

This sub is mostly just NSFW puns now.

If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.

Edit* I'm not replying any longer and turning off notifications but to all those that say "no one cares", there sure are a lot of you arguing about it. Maybe I'm wrong but you people don't need to be rude about it. If you really don't care, don't comment.

What did 0 say to 8 ?

" Nice Belt "

So What did 3 say to 8 ?

" Hey, you two stop making out "

When I got home, they were still there.

I won't be doing that today!

You take away their little brooms

This morning, my 4 year old daughter.

Daughter: I'm hungry

Me: nerves building, smile widening

Me: Hi hungry, I'm dad.

She had no idea what was going on but I finally did it.

Thank you all for listening.

There hasn't been a post all year!

I completed AoC 2020 in December 2020 and AoC 2021 in December 2021, and because I just couldn't get enough and because I've heard so much about IntCode that I wanted to see it for myself, I started AoC 2019 the day after Christmas. It's all been fairly smooth sailing until Day 16, Part 2, where I hit a bigger brick wall than I remember in either of the more recent years. After more than a day of banging my head against it, I finally have a working solution. I thought it would be worth documenting, for others who come along behind me, and so that other people can tell me where I could have saved some time and frustration. In case it's not obvious, the entire rest of this post is a spoiler. It's also extremely long and tedious.

Introduction

If you don't recall, 2019 Day 16 is the Flawed Frequency Transmission protocol. We are given a digit sequence and need to send it through 100 phases. In each phase, each digit is replaced by a new one that is computed by summing all of the existing digits multiplied by either 0, -1, 0, or 1. Specifically, when computing the new value for the first digit you use the repeating pattern <1, 0, -1, 0> for the multipliers. For the second digit you use <0, 1, 1, 0, 0, -1, -1, 0>. For the third digit you use <0, 0, 1, 1, 1, 0, 0, 0, -1, -1, -1, 0>, and so forth. The result of each phase is still a digit sequence, so we discard everything except the final digit of a sum.

In part 1 we are given a relatively short digit sequence to which we are to apply this process (mine was 651 characters long). And a straightforward implementation of the described procedure runs through the 100 phases quite quickly to show you the final output. In part 2 you are given a sequence that is 10,000 times longer, and an offset into it of the short segment you are required to type in as your answer. Now that quadratic algorithm isn't looking so good.

After a few hours of making no real progress, I searched reddit for a hint. What I discovered was lots of people saying to note that the offset you are given is always somewhere in the second half of the sequence, and that this would make your life a lot easier. [Aside: I'm one of those people who hates puzzles that want you to write code that only works for your input and not any legal input. Moving on.]

Discovery

So I started looking at what happens in the back half of a sequence. In particular, I started at the very end. In the analysis below, w, x, y, an

Buenosdillas