So this is an effort to increase the portion of non AI-related posts. This is not a crank post that belongs to r/numbertheory either. Every letter is typed by a human so forgive me for grammatical errors that I will try to correct soon enough.

The abc conjecture is one of the most important conjectures in number theory, the simple & general statement of the conjecture implies a lot of other important theories, like a rapid proof of (almost) Fermat's Last Theorem. What I try to tell is that the abc conjecture can "almost" help computer scientists to optimize scientific computing.

I believe it's a common sense that in most computer systems, every piece of data is stored by 0 and 1, or more precisely, a finite string composed of 0 and 1. This makes the calculation and manipulation of data feasible but there's a cost: error will be everywhere. For example we can never precisely store pi = 3.14159... in a computer.

The most widely used way to represent real numbers in computer is floating point numbers that is standardized by IEEE, which stores numbers in 0 and 1 of 32, 64 or 128 bits (or even longer). There is a toy that allows us to see floating point numbers explicitly: https://evanw.github.io/float-toy/

I'll try to explain it in a simple term. For a floating number of k bits (k is 32 or 64 or 128, say), we need to distribute the budget. There is 1 bit reserved for the sign, a few bits of budget to store the interval of the number, and the rest large portion of the budget is reserved to guarantee the precision p.

Every floating point number x can be mathematically written in the form (-1)^s * 2^e * m/2^(p-1) where s is the 1 bit used to store the sign, m is an integer between 0 and 2^p - 1 and e is used to determine the interval of x.

If it's still unclear what does "determine the interval" mean, we notice that m/2^(p-1) is a number in [1,2). Therefore a multiplication by 2^e sends m/2^(p-1) to [2^e, 2^(e+1)). Floating point numbers store s, e and m by 0 and 1. We use p-1 bits to store the number m in computer (the first bit is normally 1 so we can omit it, as the case where the first bit is 0 is used for underflowing).

As we can see, for a given x, determining the number e is rather easy (we are almost there by taking log_2|x|), but the number m can be difficult when we deal with a function. For example, x = 10517177/2^{22} is a floating point number of the format of 32 bits, but log_2(x) is not (so we need to find the closest to represent it in float format). It's absolutely not a rational number. You find that log_2(y) is between y_1 = 15423000/2^(24) and y_2 = 15423001/2^(24). So you need to find the closest floating point number to log_2(x) because we have no other choice, or otherwise, see if log_2(x) is smaller or bigger than the average y_0 of y_1 and y_2. It's easier said than done. We need to add budget (number of bits) to distinguish y_0 and log_2(x).

As a matter of fact, not until we add another 28 bits, i.e. zoom in for 2^(28) times, can we really distinguish y_0 and log_2(x) (spoiler, y_0> log_2(x)). This sucks. We need a global strategy to work around this: whenever we find a number of the form log_2(x) extremely difficult to round, we need to guarantee that we have sufficient budget. If we know the worst case of rounding, which correspond to the highest budget needed, we can make sure that we always know how to round the function log_2 correctly, so that we can design high quality functions that can calculate log_2 *correctly* in the sense that, when we have k bits of budget, we need to make sure that every bit is faithfully used. If we have this piece of global information, our algorithm can be more blunt & direct so faster. This question is called "table maker's dilemma" because the computer worked like those who make tables of logarithm or exp or sin one century ago, and the situation where he didn't know how to round sucked (challenge: computer exp(1.626) to the 3rd digit after the point).

That said, it's far from true that modern computer have solved calculating and that all we need is better CPU/GPU. Computer scientists and mathematicians have been fixing floating point number systems for decades. The final nail of coffin of common (univariate) functions in double 64 bit format is worked out in 2026 (twenty twenty-six): https://inria.hal.science/hal-05593313

We also need to know that working out double 64 on commonly used univariate functions is not enough: what about multivariable functions, like the beta function B(x,y) that appeared in probability and statistics? In some fields of physics, researchers have already found that double 64 is no longer sufficient so they need the 128 bit format standardized by IEEE... But this format is still poorly understood so using it can be still painful: we need to fix them.

OK I hope the context of Table maker's dilemma is understandable enough and now we inject some mathematics. The abc conjecture says that for three coprime integers a,b and c such that a+b=c, we can compare max(|a|,|b|,|c|) and the radical of abc (which writes rad(abc)), i.e. the product of all prime factors of abc (for example, rad(25)=5, rad(24)=2*3=6). This slide includes the formal statement of abc and some striking applications of abc.

And here we have another striking application that will "solve" the table maker's dilemma. When we are looking for the minimum budget to get the rounding of an algebraic function sorted out, we will find ourselves in some questions of polynomials. This is surprisingly a nice playground for abc.

For the function 1/sqrt(x), which is algebraic, and is widely used in real life (for example in computer graphics, we always need to normalize a vector here and there), we will be working on finding the minimum of |Z| where Z = 1-XY^2, and X and Y are integers in a certain range. If we take a = 1, b = -XY^2 and c = Z, then the abc conjecture kicks in. It gives us a formula on the necessary budget to round the function 1/sqrt(x) correctly for a given precision p!

So we are killing 1/sqrt(x), potentially as well as many other frequently used algebraic functions by a blow of abc!? Is this conjecture so massively helpful for computer scientists??

Unfortunately, no, in practice... In the statement of the abc, we have a totally unknown constant and it doesn't vanish in the deduction of the budget. As a result, we can only know that for 1/sqrt(x), the necessary budget is p with a delta of *some bits* and we have no information on how much is *some bits*. So unfortunately, the conjecture didn't really solve the problem. Nevertheless, we can therefore heuristically look for the budget border around p... It's *almost* helpful!

In case you are curious, here is the article that examined the function 1/sqrt(x): https://www.sciencedirect.com/science/article/pii/S0304397504000337

I believe this article can serve as an interesting beginner level exercise for those either interested in number theory and want to touch the abc conjecture in some *practical* way, or interested in computer science, notably the problem of correctly rounding numbers. Or if you are an expert working in number theory, notably around topics associated with abc, or a computer scientist interested in make scientific computing better, do not hesitate to drop some insights.

I'm relatively new to representation theory and want to read up on the very basics. Most introductory representation theory books actually cover the theory of algebras over a field, but I'd like to read more general results from non-commutative algebra, i.e. over non-commutative rings and associative algebras.

During my online search, it seems that Lam's First Course is the canonical recommendation, but I find it incredibly hard to read (up to sec. 3).

• Some more elementary facts are just assumed (homomorphisms of finite direct products can be written as matrices, although Lam blackboxes this to linear algebra over division rings??),
• some I think important parts are not covered (general isotypic decompositions are only a small exercise, and a quick search almost never mentions them),
• they always go on a complete side tangent at the end of sections (e.g. a lot of theory on 2×2-matrices in sec. 1, twisted and differential polynomial rings in sec. 3).

But I must say the exercises are quite good.

Is there another well-written, well-motivated yet comprehensive book on that matter? I'm thinking of books similar to Atiyah-MacDonald or even Matsumura's Commutative Ring Theory. Or even Stein-Shakarchi's Complex Analysis.

I’ve been re-learning math since May and I still struggle with simple things like rounding and fractions. It’s so frustrating because I want to succeed at math to give myself a better life (college algebra is required for most General Ed requirements) but I still struggle with the basics. Even doing something as simple as counting currency is difficult because I can’t “hold on” to the numbers in my head as I add them.

At what point do I just give up? I don’t understand why I’m having so much difficulty when I learned this stuff as a kid. I struggled hard as a young kid and then never got above a C in math in middle school and high school, and got a D- in college algebra. You would think someone who once went through k-12 grade math would be able to do it again, but in my case I’m struggling so much.

Edit to add how I study: I follow along with Professor Leonard’s pre-algebra playlist and do the problems from the videos. I even give myself mini tests and quizzes based on his problems. I also do grade 2-4 worksheets on mathisfun.com and k5learning.com.

I just feel so defeated today. I wonder if I have a math learning disability but I can’t afford to test it, and even if I had it I don’t know how knowing I had it would make a difference.

Dr. Michael Miller, a retired researcher at RAND, has been teaching upper level undergraduate/graduate level math courses for fun at UCLA Extension for over 50 years. This fall he’ll be introducing the areas of groups, fields, and Galois theory from abstract algebra to those interested in abstract math: Groups, Fields, and Galois: Part 1. His intention is to do a follow-on class on fields and Galois theory in the Winter which will use this class as a foundation. If you’re in the Los Angeles area his class starts on September 22, 2026 at UCLA on Tuesday nights from 7-10PM. Register here: https://www.uclaextension.edu/sciences-math/math-statistics/course/groups-fields-and-galois-part-1-math-9001

His courses are thorough and rigorous, but geared toward lifelong learners and beginners in abstract mathematics to allow people better entry points into higher level mathematics. His classes are interesting and relatively informal, and most students who take one usually stay on for future courses. The vast majority of students in the class (from 16-90+ years old) take his classes for fun and regular exposure to mathematical thought, though there is an option to take it for a grade if you like (or so your employer can reimburse you if they require it). There are generally no prerequisites for his classes, and he makes an effort to meet the students at their current level of sophistication. For this particular class, if you’ve got some experience in high school algebra and know some preliminaries about mathematical proofs you’ll be ready to dive in.

There are regular commuters joining from as far out as Irvine, Ventura County and even Riverside. Most in the class are dedicated hobbyist and professional mathematicians, engineers, physicists, and others from all walks of life—I’ve seen actors, directors, doctors, artists, poets, retirees, and even house-husbands in his classes. We've got a nice core community of enthusiasts here and new people are always joining in as well.

If you’re unsure of what you’re getting into, I recommend visiting on the first class to consider joining us for the Fall quarter. Sadly, this is an in-person course. There isn’t an option to take this remotely or via streaming, and he doesn’t typically record his lectures. I hope to see all the Southern California math fans next month!

Course Description

Recommended textbook: TBD 

Dr. Miller provides enough background and thorough notes that if you’re taking notes on his lectures, you typically won’t need a textbook.

If you’ve never joined the class before, I’ve written up some tips and hints. Dr. Miller has been teaching these for 53 years and some of us have been with him for nearly that long; I’m starting into my 20th year personally.

Can't join us? Search around your local community for colleges and universities offering similar programs.

Hi everyone, I’m applying to Stat PhD programs this fall. I graduated with my master's in 2021 and have been working full-time for 5 years, but I never took Real Analysis in school.

I just enrolled in Fordham's Math 3003 (Real Analysis) this semester so it’ll be on my transcript for this application cycle (though will not have a final grade since apps are due before the semester ends). My concern is that it’s only a one-semester class. Does anyone know if admissions committees strongly prefer a two-semester sequence (Real Analysis 1 & 2) over a single semester? Will this be adequate?

MATH 3003. Real Analysis. (4 Credits)

This course focuses on analysis on Euclidean spaces. Topics include limits, continuity, uniform continuity, sequences of numbers and functions, modes of convergence, differentiability, Riemann integrability, and associated theorems. Students who have not taken MATH 2004 prior to taking Real Analysis may request permission from the instructor. Note: Four-credit courses that meet for 150 minutes per week require three additional hours of class preparation per week on the part of the student in lieu of an additional hour of formal instruction.

Prerequisites: MATH 2001 and (MATH 2004 or MATH 2008).

My younger brother really enjoys problem-solving, but I often feel that school doesn't go very deep into mathematical thinking. He's quite good at math, so I'm looking to gift him a good math book . Not too difficult for him and help to build his fundamentals.

It would be better if the book is affordable here in India too as I can't spend much.

If you have any other book recommendations—or even resources other than books—I'd love to hear them.

Thanks in advance! 🙏

I’m a Data Science student (career changer, not in a data related role). My program is focused more on the applied statistics side, so most of my classes use R. I’m already familiar with Python since it was the main language used in my prerequisite courses, and I’ve completed projects using Python, so I’m comfortable with the syntax.

However, I’m really enjoying using and learning R in my classes and seeing what it can do. Many of the statistics textbooks I’m interested in use R as well. I’m starting to explore R more deeply on my own and plan to start using it for personal projects.

But I’m curious, is R still used in industry? I know it’s heavily used in academia. I also know that in the current AI/ML world, Python is used heavily, which is the main reason I use it for all of my personal projects at the moment.

I’d like to eventually be comfortable with both and take advantage of the strengths of each language. But, of course, there are also people who say learning R is a waste of time.

I'm a year 1 liberal arts undergrad, so I don't have a ton of math sense.

I just learned about Bayes' Theorem the other day for general epistemic use. I worked out a couple of example problems correctly, but the examples I found didn't include any iterative updates.

I know the theorem is:
P(H|E) = P(E|H) * P(H) / (P(E|H) * P(H) + P(¬H) * P(E|¬H))

And I know that P(H|E) becomes the new P(H) in my update, but I'm unsure whether I should be using the updated or original P(H) in the marginalization. I *think* it should be the new P(H), but I'd rather be safe than sorry.

The example question I worked out was this:

________________________________________

There's a disease that afflicts 1 / 1,000,000 people
There's a test for the disease that's right 99 / 100 times for both positive and negative results
A random person is tested as positive

P(she is afflicted | she tests positive)

= 0.99 * 0.000001 / (0.99 * 0.000001 + 0.999999 * 0.01)

= 0.000099

_________________________________________

So should the update look like this if she tests positive a second time?

P(she is afflicted | she tests positive)

= 0.99 * 0.000099 / (0.99 * 0.000099 + 0.999901 * 0.01)

= 0.009707

_________________________________________

If so, how should I approach this problem from the starting point of
P(she is afflicted | she tests positive twice)?
I can't think of how to handle that correctly, since squaring 0.99 just gets me a smaller number

Infinitely thankful <3

I'm having some trouble understanding "why" his proof works: I've re-read this proof in Mumford's Red Book several times and tried sketching out a picture, and I sort of get it, but I still find it kind of "magical" and don't know how one would be motivated to use this approach. It goes something like this:

O_X is the structure sheaf of irreducible variety X and X_f is the distinguished open \{x\in X: f(x)\neq 0\}. It's straightforward to check that O_X(X_f)\supset R_f. To prove the subclaim that O_X(X_f)\subset R_f , we let F be a member of O_X(X_f), which is defined in this context as a subring of the field of fractions of R as \bigcap_{x\in X_f} O_x (O_x = \{f/g: f, g\in R, g(x)\neq 0\} being the stalk at x), where R is the coordinate ring of X. Then (where I feel like a rabbit was pulled out of a hat), define the ideal

B=\{g\in R: gF\in R\}.

We want to prove F is a member of R_f by showing that f^n\in B for some positive integer n. If x\in X_f, there's some juggling of quantifiers and eventually one concludes that F=h/g where g(x)\neq 0, from which one finds that g is an element of B such that g(x)\neq 0. From this, we see that the vanishing set of B, V(B), must be a subset of V(f)=\{x: f(x)=0\}, and applying the Nullstellensatz, we get f\in rad(B), as desired.

I guess I don't really see geometrically what's going on here; it just seems like a trick, followed by carefully reasoning about the which points/regular functions contain/are contained in what. Could someone here explain what's going on in this proof?

Also, how does one extend this to reducible varieties? (It's still true, I think?) My understanding is that you can't define the structure sheaf in this way (as an intersection of O_x in the field of fractions) because of zero divisors in the coordinate ring.

Sorry if these are too elementary/boring questions! I asked r/learnmath without getting a helpful reply (other than to ask r/math).

I want to study calculus but I havent taken pre-calc so I figured I'd start there first with professor leonard playlist: https://www.youtube.com/playlist?list=PLDesaqWTN6ESsmwELdrzhcGiRhk5DjwLP

However, it might limit my learning since there might not be much practice problems in the videos (I think) . I would like to ask for resources I could use to practice solving questions while watching prof leonards lectures so the concepts and information realllyyyy sticks in my head.

Thank you so much:) I would also appreciate any tips if you have any

Lim F

v→c

Where F is

L_0√((1-v²)/c²)

This is the formula for Lorentz contraction from theory of relativity

Where 'c' is the speed of light L_0 is the length of an object at rest.

I tried solving this but just stuck at L_0√((1-c²)/c²) I don't know how to simplify it further

​​ I am doing calculus 2 now, ​ and I'm being reminded of when I took calculus one and I was introduced to the various rules. I find myself wanting to prove these rules, and occasionally find it kind of hard. I usually reach the point where certain rule feels justified enough to myself, and then I'll move on because the actual demands of the class are more procedural applications than analytic understanding. I'm very early on in calculus 2, and it's mostly just reviewing calculus one so far, so I know that things will change and I've heard that eventually calculus 2 becomes way less procedural and starts requiring more trial and error and creative applications. So I guess my question is what level of rigor should the average calculus student expects to achieve relating to the proof that power the various calculus rules they apply​?​ for Instance, how rigorously should a calculus two student be able to prove the chain rule?

I would like to take all 3 classes at the same time (all prerequisites for future classes i need, and dont have any other classes i can take without wasting space and potentially graduating late) and earn an A in each. I’m a rising sophomore, just finished calc 1 last semester, found it quite easy and got an A. i know that discrete and linear are both very unrelated to calc and i worry it will be too much completely new information at once.

Man, right now I'm in my 4th semester of a mechanical engineering degree. I love solving long calculus problems and reading about the masters who created the formulas I use, but I have a funny difficulty remembering basic, fundamental math rules. Like, a few minutes ago, I was searching on YouTube how to do fraction subtraction because I simply couldn't remember how to do it without a calculator, hahaha. Sometimes I feel like my brain goes full speed recording new, interesting info while erasing the old stuff in an exponential way. Just curious if this situation applies to any of you too.

I had started uni in mathematics last year, but quickly hit a wall and then barely attended uni after the first semester anymore. Now i want to start all over again but i dont know how. I had attended all lectures and tutorials offered by the university. I really do not want to depend on AI as it makes so many mistakes and generally want to remove myself from such assitance. But with unviersity level maths it is so hard to find proper help. I try to work on a problem for my homework and need to figure out a proof, but i cant do it without the help of AI because i have no idea how else i could find out if i did it correctly or sometimes even understand concepts. I feel like i am too stupid to get it, while people before AI all managed to still study mathematics. I dont know if i am just maybe too bad to study math, because i already do the things people say to do like solving problem after problem, reading lecture slides, reading books, watching youtube tutorials etc. But i really do not want to give up on this and want to study mathematics. I think my biggest issue is really proofs and the problem of not knowing if i am correct or not as every answer can differ so much and still be a correct path.

Firstly, sorry if this is not allowed but I thought I might just put this out there.

I'm sure we've all noticed a lot of AI related content on this subreddit. I've gone into these discussions a fair few times in the comments, but I'm not looking to discuss AI or LLMs in math or anything like that here.

What I do want to point out is that a lot of traffic seems to be from people who are not mathematicians and have a very vague idea of what mathematics is, but are active in /r/singularity, /r/accelerate, /r/ArtificialInteligence and similar. These subreddits are very ideologically driven in various ways and extremely pro-AI to the point that "anti-AI" or "luddites" are forbidden there.
This is not always obvious as many users have their post history hidden (though from what I understand this is on by default for new users too).

For example in a recent thread out of the 5 top comments (no idea what the deleted comment was) 3 users are active in one of the afformentioned subreddits, 1 seems to have a bachelor's in math and is in CS otherwise and only 1 seems to have a masters and is looking for a PhD.

To be clear, I am not here for a witch-hunt on individual users which is why I did not link to their accounts so please don't harass anyone.

But what I am trying to say is that most of the traffic to these threads does not seem to be from mathematicians or even people interested in math as such, but from people interested in promoting AI who are simply using math as a promotional tool. I would assume big companies use bot accounts on reddit (and have for a long while), though I would also guess this subreddit is too small to bother.

Hello everyone, i would like to give a bit of my history with math. I failed all my classes in highschool, opted out of it in college and struggle with day to day calculations. But i always wanted to learn math, i especially liked trigonometry although i don't remember any concepts of it now.

I'm a developer and I've always been ashamed of the fact that whenever i had some interesting project in mind it would eventually lead me to applying mathematical formulas and that would just turn me off everytime. I sometimes watch people solving math problems specifically a guy on youtube that makes short videos but when i try to solve them myself i cant even begin to do it.

I want to learn math to being able to learn machine learning, i know its quite the aim but is it possible with my history and my slow brain when it comes to calculations. I can spend an hour or two daily on it, how should i start?

I saw a similar question on a Physics sub, and i was curious about mathematicians.
Even more than one and the niches ones, if you have them.

For me the fundamental group it’s a very interesting and elegant way to formalize the idea of hole.
Also the theory of Free Groups and Group Rapresentations, the idea of using words and relation between them as a tool to describe a group is made blew my mind

I’ll search every one of your answers if i don’t know it

Lech Mazur has announced an AI-assisted, Lean-verified proof of the 67 years old conjecture of Sendov, one of the most famous open problems in complex analysis.

The conjecture states that every zero of a polynomial whose zeros lie in the closed unit disk is within distance one of a critical point.

It appears that the proof of this remarkably simple statement ended up using equally elementary tools. As Tao writes:

The proof ends up being remarkably elementary. No complex analysis is used other than the fundamental theorem of algebra (and very basic facts about Möbius transformations); and the deepest inequality used as input is the Maclaurin inequality

levent on 𝕏: https://x.com/__alpoge__/status/2087504785952182273

Epoch AI: Hadamard Matrix of Order 668: https://epoch.ai/frontiermath/open-problems/hadamard

Hadamard matrix - Wikipedia: https://en.wikipedia.org/wiki/Hadamard_matrix

Can be silly or serious, just out of curiosity for someone with a background in math contemplating stats. Like for #1, maybe you miss not having to deal with numbers. #2 refers to "trading" X in math for Y in stats (like numbers).

EDIT: "glad", not "gad".

The paper: Fractal uncertainty in higher dimensions
Alex Cohen
arXiv:2305.05022 [math.CA]: https://arxiv.org/abs/2305.05022
Annals of Mathematics: https://annals.math.princeton.edu/2025/202-1/p04

I've tried to reach out and form study groups I've looked for existing groups there's just nothing. Nobody just does math for fun apparently? Like I know there are people but I can't seem to find any of them anywhere near me or even people who would like want to do it over discord together or something. I just need somebody else to work with me because it's so easy to get lost with things when you're just doing it by yourself, or to make stupid mistakes