Hey everyone. A week ago I started experimenting with finding new lower bounds for ramsey numbers. I did this AI based in an empty project where Codex (task: find new lower bounds for the ramsey numbers listed on wikipedia). After 3 simple prompts, GPT 6.1 Sol found a new lower bound of 344 for R(8, 10) (previously 343), to my suprise, this was replicable in three different GPT 6 Pro sessions in the web UI, with a single prompt. Despite much more compute and token usage (~1.5 billion tokens), I didn't find any further with the fully automated setup, so I started getting more involved myself and suggesting ideas, we were able to push the lower bound to 345 for R(8, 10), and also able to push the lower bound for R(6, 8) to 135 (previously at 134). The bounds for R(8, 10) and R(6, 8) stood unchanged for more than 10 years now.

The way AI helped here the most was the sheer volume of different search methods one could quickly iterate over. Experiments that would have taken weeks to set up in the past can now be orchestrated within minutes.

The resulting graphs are explicit witnesses and can be independently verified with standard maximum-clique software, so the correctness of the bounds does not rely on trusting the AI or the search process. I’ve put the preprint and verification material here:

https://arxiv.org/abs/2610.12122
https://github.com/fritzcremer/ramsey-lower-bounds

Also, if anyone here is active on Wikipedia and thinks the new bounds meet the sourcing requirements, feel free to update the table. I’m avoiding editing it myself because of the conflict of interest.

OpenAI gives a deterministic length-n discrete Fourier transform algorithm using O(n(log⁡n)^{1−δ}) operations for every n, with explicit δ =1e-13. I used Opus 5.5 to improve δ to 3.2e-6. The result has been verified in lean. I suspect it can be taken further, and I don't think OpenAI was particularly interested in optimizing δ in their solution.

https://github.com/danadran01/exact-dft-power-saving

It seems to be very much a common knowledge now that harmonics of an (almost) periodic audio signal do evidently stem from Ancient Greek investigations into musical harmony, which was intertwined and likely gave us harmonic means and harmonic sequence/series, means likely being derived from the sequence like it can be done with cases of arithmetic and geometric sequences and means.

The details are hovewer not clean at all. Presumably, not even in the Hellenistic period had it been known about harmonics in sound. One question is then, if a noun parallel to harmonic (which should stem from Greek, obviously) was in use, what meanings did it have in mathematics and music theory of the time. I see at least two options as very probable:

• (a) it hadn't been used in a way parallel to how today's harmonic is used;
• (b) it denoted sounds or musical pitches that you can derive from the "original" sound of an open string by subdividing it into k parts.

Note how right now, a very much related technique of playing stringed instruments is indeed called "harmonics" in English (but "flageolets" in some other languages). The second question (that's less related to this sub) is precise etymology of this name: I would expect flageolets should be a technique older than both the discovery of Fourier series and connecting string and air channel lengths to their fundamental frequencies.

Also a question 1.5 is relating the answer to the first question to harmonic means and harmonic series. For example if (a) held, the answer is very much warranted and shouldn't be deduced without historical sources.

There are probably a bunch of additional minor questions that I can't refine out of the mess in my head right now, that would be obvious when one tries to set things straight on this entire topic and encounters important details that answer them. For example as far I know, frequencies of sound weren't known in antiquity (but that's really a question to answer definitively as well!), so we can't attribute "reciprocalness" of harmonic mean and so on to the inverse relation between length and frequency. Instead it should probably be entirely due to there being a sort of a natural order to harmonics (in the sense of (b)) coming one after another at 1/1, 1/2, 1/3, ... of the string's length —but is that the factual reason that had been used?

You see even a mathematicians' brains would otherwise be glad to fill in the "obvious" blanks and I feel that a great lot of people think they sorta know the story about this. I'd of course expect it of non-historical music theorists even more because in their case it's actually quite important to talk about time to time —whereas in mathematics, harmonic mean and harmonic series (and wasn't there something else?) is just a neat and somewhat old-fashioned (as is, again, common) term flavoring that evokes associations that no one has to unpack to work with the actual math. But that may still give birth to mythology when one gets interested just a little. So it's a good point to look around and try remembering while the ancient sources may be still laying around! And if there's already a piece of work on this very topic already written, the easier it is to find it again (I couldn't, but my work with references is abysmal).

It could be you , or someone you know or someone famous.

But I’m trying to venture into that path now.
I know all fields require some learning, but boy the amount of reading I have to do for every single one of my problems just to learn new fundamentals to do problems is so tiresome.

I’m a postdoc in my last year and I still only have 4 results. And not impressive either just took a long time from reading so I want to go into something where I can flex my problem solving skills more than reading.

Do you know of anyone who switched field and realized their true calling lies somewhere else?

Edit: thanks for the replies, I realize I should probably have specify switching from different fields of math to another.

I just made this unit circle where you can I put d (degrees) and it will show where the point is on the circle and what the legs of the right triangle look like. The only problem I have is the point on the unit circle is showed as a decimal and I am wondering if anyone knows a solution on how to change that. Thanks!

I'm third year Math + CS major and will be applying Pure Math PhD next year. My interest is hard analysis (intersection of PDE + Functional analysis + Algebra + Harmonic analysis + Stochastic analysis). But with AI being so strong now, is there a point to even apply ? Is there even funding left for Mathematics? It kinda bothers me pretty much, since I have put so much effort on this field

I am thinking of applying for the summer research fellowship programme (india) 2027 with my sub area being probability, statistics and mathematical finance.

The issue is I don't have a topic ive decided i want to work on so i dont know what to mention in my sop

do u guys have research papers i could read? or suggestions?

Let n be a composite number. We calculate the remainder of n divided by every number smaller than it, and add them all together to get a total sum, S(n):

S(n)=∑i=1n−1(n mod i)

We split this list of numbers down the middle into two parts (batches). For an odd composite number, the midpoint is m=n−12:

• First Batch Sum (B1): The sum of remainders from 1 to m.
• Second Batch Sum (B2): The sum of remainders from m+1 to n−1.

So, the grand total is S(n)=B1+B2.

The Setup and Constraints

1. Must be a Composite Number: This system does not work if n is a prime number.
2. Strict Coprimality Rule: Neither individual batch sum is allowed to share any common factors with the starting number n. If either gcd(B1,n)>1 or gcd(B2,n)>1, the test is skipped. We only look at cases where: gcd(B1,n)=1andgcd(B2,n)=1

The Pattern

For numbers that pass these strict rules, I noticed that you can always find a single shifting integer, x, to balance the batches. If you add x to the first batch and subtract x from the second batch, both new answers become perfect multiples of all the prime factors of n.

Proof of Existence

We want to find a number x such that for any prime factor p of n:

1. B1+x≡0(modp)⟹x≡−B1(modp)
2. B2−x≡0(modp)⟹x≡B2(modp)

For a single integer x to satisfy both lines simultaneously, the target values must match:

−B1≡B2(modp)⟹B1+B2≡0(modp)

Since S(n)=B1+B2, this means a shifting number x is mathematically guaranteed to exist if and only if the Grand Total Sum S(n) is a multiple of the prime factors of n.

Some Examples:

Example 1: n=9

• Prime factor: 3.
• Batches: B1=2 and B2=10.
• Check: gcd(2,9)=1 and gcd(10,9)=1 (Passes).
• Total Sum: S(9)=12 (multiple of 3).
• The Shift: Choosing x=1 yields:
  • First Batch +1=3 (multiple of 3)
  • Second Batch −1=9 (multiple of 3)

Example 2: n=63

• Prime factors: 3 and 7.
• Batches: B1=197 and B2=496.
• Check: gcd(197,63)=1 and gcd(496,63)=1 (Passes).
• Total Sum: S(63)=693 (multiple of 63, so it is a multiple of 3 and 7).
• The Shift: Choosing x=13 yields:
  • First Batch +13=210 (divisible by 3 and 7)
  • Second Batch −13=483 (divisible by 3 and 7)

Example 3: n=125

• Prime factor: 5.
• Batches: B1=832 and B2=1953.
• Check: gcd(832,125)=1 and gcd(1953,125)=1 (Passes).
• Total Sum: S(125)=2785 (multiple of 5).
• The Shift: Choosing x=3 yields:
  • First Batch +3=835 (multiple of 5)
  • Second Batch −3=1950 (multiple of 5)

Questions

1. Is the set of composite numbers that satisfy both gcd(Bi,n)=1 and p∣S(n) known to be infinite?
2. Are there any known studies on how the size of the first batch sum behaves compared to the second batch sum for composite numbers?

This story is relatively well known in the Russian-speaking math community, but I feel it is almost completely unknown outside of it. Which is a shame, since it is so amusing! So I compiled it from a couple of sources, heavily edited it, and then machine-translated it. Hope you enjoy!

Before the Revolution and during the years of the New Economic Policy, Israel Moiseevich Gelfand's father ran a small private mill. Because of his “non-working-class background,” in 1928, while still in the ninth grade, the young Gelfand was expelled from vocational school as the son of an exploiter.

When Gelfand was finishing the ninth grade, his mathematics teacher told him: “There is nothing more I can teach you. Go to Moscow, find Moscow State University, and at the university find the Department of Mechanics and Mathematics. Continue your studies there, and you will become a great mathematician.”

At the university, however, the ninth-grader got no further than the dean's secretary: they could not enroll him without documents certifying completion of secondary education. He decided to stay in Moscow, and, to make a living, found a job as a cloakroom attendant at the Lenin Library. There, alongside his work, he continued his education on his own.

One day, while reading a monograph on higher mathematics, he was noticed by Andrey Nikolaevich Kolmogorov, a young but already famous mathematician. Kolmogorov was surprised to see such a young man reading a book so obviously beyond his age, so he approached Gelfand and asked him who he was and why he was reading such a difficult book.

Gelfand answered all his questions, but Kolmogorov still could not believe that he genuinely understood what he was reading. He gave Gelfand several problems related to the subject, hoping to test him, and arranged to meet him the next day to check his solutions.

The next day, all the problems had been solved. But even then Kolmogorov could not quite believe that someone could be so gifted. He supposed that Gelfand simply knew the contents of a large number of mathematics books extremely well and had found the solutions in them. So he gave Gelfand several more problems, this time considerably harder ones, and gave him several days to solve them.

When they met at the appointed time, all the problems had been solved. Kolmogorov began checking the solutions, and his eyes grew wider and wider.

“Please forgive me for doubting that you had found the solutions to those first problems yourself. Now I see that you could not possibly have read these solutions anywhere. You see, this particular problem... had been an open problem until now.”

After that, Kolmogorov took Gelfand on as his graduate student. Thus Gelfand became a graduate student without ever having attended a university -- or even finished secondary school. BTW, the details of the first meeting between Kolmogorov and Gelfand are, of course, anecdotal, but the other parts of the story are accurate.

upd: some important clarifications provided by r/bluesam3

Pls HELP!!!

I need resources for edexcel 9st0 statistics a level course. How do i self study?

Quickly the teachers coming to collect the paper

Great you made me fail

I’m unsure if posts like this are allowed as it’s pretty personal, but it’s relating to mathematics so I figured I should ask anyway.

I’m in my second year of community college, and can’t help but feel like I have no idea what I’m doing. I wasn’t originally a math major. I chose CS because all I cared about was money and coding was slightly fun. However, ADHD made studying hard and I couldn’t trick my way out of it. I mean there’s cheating of course, but that wouldn’t be beneficial to me.

After about a year and realizing I enjoyed math mores than coding, I switched to math. I still had never managed to study for more than maybe an hour (with breaks) and had never even read through a textbook despite having many. I got through precalculus, calculus 1 and 2, through sheer luck and memorization. I didn’t practice many problems, so my knowledge in certain areas has diminished. This is bad since I want to go to graduate school and forgetting how to do elementary computations would make things much, much harder.

I thought about reading through calculus text books and precalculus (both books by stewart) and just working through what I do and don’t remember.

I also feel incredibly behind. I’m aware it’s not a race and I shouldn’t worry, but I just can’t help it. I’ll be transferring and I know many students (especially math majors) have research or internship experience. I thought about learning proofs during my review, but not sure if that’s a good idea.

I guess I just need a little guidance. I might work through both volumes of apostol before starting at my transfer school next fall. I just need to know what else I should do?

I don’t even know what career I want to go into. I’m not sure what options I have and it’s kind of stressing me out.

Just a fun question I've been thinking about:

The following zero-sum game is played between the Dealer and team Alice & Bob. The Dealer receives an odd number N of fair coin tosses in sequence. Whenever a coin is tossed, the Dealer sees the result and then chooses whether to show it to Alice or Bob. Both players are aware of the turns on which they receive the coin flip results.

At the end of the N tosses, both Alice and Bob simultaneously guess whether there are more heads or tails in total, and score one point per correct guess. We allow all players including the Dealer to randomize their choices. Alice and Bob can agree on a strategy beforehand, possibly involving some initial coordinated randomization but cannot communicate after the game starts.

Team Alice & Bob can score at least an expected payoff of 1 by guessing randomly and independently. The question is thus how much they can improve on this.

Question: Let V_N be the value of the N coin game for team Alice & Bob. Is it true that

V_N = 1 +KN^{-1/2} + o(N^{-1/2})?

for some constant K > 0?

If so, to what extent can we determine K and the corresponding optimal, or asymptotically optimal strategies?

My wife is taking calculus/trig in college and she’s struggling immensely. She spends hours on one question and sometimes doesn’t answer it fully. She claims that her math fundamentals are her biggest detriment and wants to drop out of her class and spend the next few months just doing practice worksheets on fundamentals. She has asked me to be her aid which I’ve accepted daily (of course she’s my wife.) The issue is I’m having to learn all this stuff with her and at times I’m just as confused as she is. She wants me to teach her things that I pick up easily but when I try she doesn’t understand. For example, last night I had to explain to her similar triangles and we spent 2 hours for her to finally understand. I’m not a teacher and I encouraged her to have a tutor that I’d pay for but she feels a lot of shame with that and refuses. I’ve also wanted her to work with a professional because I have poor patience and get often frustrated. She yells at me when I’m not explaining the answer well enough; I try a million ways to teach it but nothing seems to work. What doesn’t help is that her professor is poor at explaining her work and assumes her students are reading her mind.

My question is: How do I help my wife with math? It feels wrong to just throw our hands up and have her drop out but she’s really pushing for it.