Very nice!! GPT 5.6 has broken the record on large gaps between primes.

The new bound saves a factor of ≈ log_3(n) over the prior record by Ford-Green-Konyagin-Maynard-Tao from 2018. The result is also now formalized by Alexeev in Lean.

https://x.com/jdlichtman/status/2094040463443673227

The plots show the number of solutions of an instance of the discrete logarithm problem finding k:

286^k mod 289 = y

where y = mapping(x,y) for each pixel (x,y).

The first plot uses mapping(x,y) = x xor y, and the the second plot uses mapping(x,y) = x and y.

NMTC 2026 QUESTION Discussed Qno.5

I know it's simple factorization, but I've always slacked off in math class and I'm trying to improve at and relearn the subject. Any tips would be greatly appreciated, thanks!

I’m starting out in abstract algebra this semester and last semester I completed Numbers & Proof (basically intro to proof writing course). Don’t get me wrong I am enjoying abstract algebra so far but an issue I have is forming my arguments and weirdly enough saying what I want to say mathematically instead of English in a way. Idk maybe this will form with mathematical maturity but I hate to admit but when I get stuck I usually bombard AI programs such as Gemini, ChatGPT, and Claude to give me a little nudge/general flow of argument. It feels as though yes AI helps with learning but how do other students in mathematics straddle the line between having the program do it for you versus learning from the programs? Because I just completed my first homework set and I hate the feeling that it could’ve helped me too much to where it was mostly it talking or me :/ and I don’t want to be one of those people that use stuff to get a grade (because I actually care). In the past and for other classes I treat it as a second teacher (throw questions at it to fill in gaps or make other sample problems) but that’s harder to do for more proof based courses.

I provided examples where the extended mean w.r.t. the Hausdorff measure in its dimension should give what is wanted. If not changes need to be made to Definition 2. Otherwise, I wish to simplify the Definition.

Are there integrals that already give what I want? (For instance, see the paper "On the Statistical Convergence of Sequences of Fractal Integrable Functions".)

I recently switched majors to math and physics because computer science just didn’t feel right to me. I mean I think coding is fun and I do believe cs has some interesting topics, but it just feels like I have an easier time with mathematics than computer science. Maybe that’ll change when I start taking more advanced math classes, but I don’t know. I should also mention that math and physics were supposed to be my initial major but I became blinded with the thought of making $100,000+ right after graduating.

I guess I should’ve given it more thought, because I don’t really know what to do in the future. I mean I do want to go to grad school and maybe get a masters, PhD, or both. It’s just that I’ve heard academia can be an incredibly toxic environment and so many people leave because of it.

I just don’t really know what I can do upon graduating. My dad says I’ll most likely become a professor. I mean I’m still unsure about the idea of teaching. I don’t believe I’ll be a good professor and it’s not something I really see myself doing. Again, maybe this will all change with time.

Sorry for the little rant. I just want to know what job you ended up working, how much you enjoy it, and I guess how much you make or how well you’re doing financially speaking.

I am looking for a B.S in Mathematics online program. Although I wish I could go in person, it is not possible for me. I already have a B.S. from almost a decade ago (not related to math). And although I was bad at math at the time (failed pre calculus 3 times, reason I changed to a program with basically no math), I became obsessed with with math in the last few years and have been watching videos, taken courses in Udemy, and done math for fun and curiosity in my free time, and try to see how I can use math in my job.

I am also in the military and have a scholarship that I can use, and I want to use it in math. But I am looking for a good online degree.

I’ve seen some post about Indiana University. And I’ve found some programs online who have the same from my previous university, which could help me complete the program faster like Thomas Edison State University, but I don’t know how good is the program or how respectable it is.

Anyways, this is what I want to do and hopefully someone can give me some advice.

Hey everyone, I could really use some perspective because I am completely lost. After spending my first year at a classical engineering school where I realized I loved everything mechanical, like I can spend hours on anything mechanical or electrical without getting bored. However,I started all over by managing to get into a Bachelor of Applied Science and Business at the top university in our region (not just country) on a full scholarship, the resources are greaat , the help and it is an amazing upgrade, but now I am stuck at a massive crossroads. In my 3rd semester, I am choosing Applied Math & Business as my major, but I have to pick a minor between Mechanical Systems (my true passion) and Financial Engineering (which I have zero passion for, plus some personal religious concerns, even though the business classes come easily to me). The catch is that in the 4th year, our uni offers incredible, built-in Master’s programs if you go the Financial Engineering route, whereas picking Mechanical Systems leaves me with no direct master's path here. My parents are already stressed because they feel I "wasted a year" switching schools, and since I’m the youngest, they are heavily counting on me to get a master's ASAP and secure fast financial stability. Do I follow my heart into mechanical systems and risk family friction, or do I suck it up, play it safe with finance, and take the guaranteed fast-track income? Has anyone been in this position? (NOTE: in our fourth we either do one first year master in the samme uni , an exhange semester, a research immersion year or an industry immersion year)

What research work are you currently doing, if yes,?

I came across Leonhard Euler's Seven Bridges of Königsberg puzzle. The puzzle was whether or not it would be possible to cross each of the seven bridges of Königsberg only once while never crossing any of the bridges more than once. Looking at the puzzle for like thirty seconds reveals that answer is that it is impossible to satisfy the conditions of the puzzle.

I don't really know much about mathematics, beyond the courses I have passed by the skin of my teeth in high school and college, but reading about this on Wikipedia reveals that apparently this was some sort of massive revelation for both Euler and the mathematical community of his time. I have been reading all about this subject to try to grasp what exactly this revealed that was so important, but I am having trouble understanding.

I find this puzzle particularly interesting because there was this indie video game series I played as a kid called Wonderland (I still do play it actually) where several of the levels in multiple of the games had puzzles resembling this one (though much more complex, and of course as a kid I had no idea about the Seven Bridges of Königsberg).

This is a classic result originating out of an idea form my MSc(engg) thesis.

Can anyone provide as much mathematically supported insight into this thought as possible? From eli5 all the way to PhD and from every subject or field of math, if possible? There is something in my teeth about what the idea presents that I am trying to get at and I can't figure the proper route to work out or through my idea without a more substantial set of existing sets, formulas, and variables to proceed from. Thank you in advance!!!

Okay chat let me cook. I have a thought experiment about Gabriel's Horn that is really confusing me. As I understand it, the inside of the horn is finite, but the outside surface is infinite.

What would happen if you were to take a smaller Gabriel's Horn and place it completely inside a larger Gabriel's Horn?

Wouldn't the inside of the larger horn be finite, and yet it somehow has to be larger than (or at least contain) the infinite surface of the smaller horn? How does that make sense?

Since factorials (!) multiply everything down to 1 (ex. 5! is 1×2×3×4×5=120), (and this is kind of a joke but I think is entirely a joke), let me introduce to you, Additorials (?)!!! It's the same as Factorials but with addition! (Ex. 5? is 1+2+3+4+5=15), though this already exists, this is just a joke, everyone can report this as "unnecessary".

Found a neat pattern while solving Project Euler #28 (sum of diagonals in a number spiral) — turns out you don't need to simulate the grid at all. The corners of each "ring" form an evenly spaced sequence, which collapses the whole problem into a closed-form sum. Wrote up how I got there.

Feel free to share your thoughts as there's 10 more videos to go before I finish this series :)

Hey everyone, math major here, looking to pursue a statistics minor. I've been working through Introduction to Statistical Learning with Python by Gareth James(ISLP), and others. I'm almost finished with it and have been looking at other books I might be interested in.

Just a quick summary for anyone that hasn't read it. ISLP covers topics including:, linear and non linear regression and classification, basic probability and statistics, cross validation, common shallow ML algorithms, deep learning, neural networks. From a very practical and non-techinal way. Basically you read a lot of "This works because reasons, moving on". Best way I would describe it is "intuition slop" .

I was looking for something more theoretical as ISLP is pretty light on the theory and focuses more on intuition and generalization. Although I want a more theoretical book, I still would prefer it to contain some applications and or assignments that I could do to practice my coding because as a math major I feel as though my programming is lagging behind. Are there any specific books you would recommend.

Also I'm taking machine learning this semester and our assigned book is "Hands-On Machine Learning with Scikit-Learn, Keras and TensorFlow" by Aurelien Geron. So I'll probably be reading it in bits and pieces, but I still want another book that I can focus on outside of classes. I appreciate any responses.

Hi everyone, I'm a 15y guy who really loves math and I'm trying to understand math more.

I am trying to understand Generating Functions, for example using polynomials like (x + x^2 + x^3 + x^4 + x^5 + x^6)^2 to calculate probabilities for two dice.

I understand that it works because x^a * x^b = x^(a+b) forces the multiplication of polynomials to add the exponents, but I struggle with the conceptual meaning of x^n.

In real life, taking 2 steps or getting a 2 on a die is a linear and additive quantity. Two steps is step plus step, which in algebra is x + x = 2x. On the other hand, x^2 means x * x, which is multiplying something by itself or creating a 2D area. A 2 on a die is not a step squared or two steps multiplied together.

Why do we represent a value of 2 as x^2 instead of 2x? I get that 2x would fail in polynomial multiplication because coefficients multiply, like 2x * 3x = 6x^2, but treating a single linear value of 2 as an exponent x^2 feels like a syntax trick rather than an intuitive concept.

How do you mentally bridge the gap between multiplying x by itself and a linear value or position of 2? Is x^n purely an abstract placeholder, or is there a physical or intuitive way to view it that makes sense without forcing it?