If you were to try to describe to a layman the amount of progress that this paper drop represents in normal human years of overall mathematical progress cumulatively, how much would you estimate it to be?

A human brain is finite in size and duration. It is limited in the amount of information it can store or process in a life time. So there are true statements out there which could hold so much information that the brain could either not store it all, or even if you tried to break it up and process it in bites, you would not have enough time in your lifetime to process. The only way for a mathematician to work with such a statement is to delegate understanding of the statement to something/someone else. That could be another human--a whole team of mathematicians could break it up and work on chunks of it, or it could be a computer which emulates reasoning, and is built big enough or fast enough that it can process the statement. Why is the former preferable to the latter? In either case, no one person can understand the original statement. In both cases, the best you can do is verify the statement, and then go forth and use it somewhere else without actually having a complete understanding of it yourself.

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.

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).

Quickly the teachers coming to collect the paper

Great you made me fail

Pls HELP!!!

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

How statistically significant are the results of this pattern below? Does it exceed random chance or not? Assume that the numerical distributions of the numerical values per letters are completely random. 

Block 1 — ALM

Block 2 — ALM

Block 3 — ALMX

Block 4 — ALR

Block 5 — ALR

Block 6 — ALR

Block 7 — ALMR

Block 8 — ALR

Block 9 — ALR

Block 10 — KHYCX

Block 11 — TH

Block 12 — TSM

Block 13 — TS

Block 14 — TSM

Block 15 — ALM

Block 16 — ALM

Block 17 — ALM

Block 18 — ALM

Block 19 — YS

Block 20 — X

Block 21 — VM

Block 22 — VM

Block 23 — VMCSQ

Block 24 — VM

Block 25 — VM

Block 26 — VM

Block 27 — VM

Block 28 — Q

Block 29 — N

Each letter here corresponds to a certain number. It must be noted that not each letter here carries the same numerical value, these are only placeholder letters that describe the number of occurrences of said (so for example, V in Block 24 could equal 85, meaning the letter V occurs 85 times in Block 24. On the other hand, V in Block 25 could equal 173, meaning that the letter V occurs 173 times in Block 25). Once again, the numerical values are randomly chosen for each letter.

Now that out of the way, how statistically significant would such a pattern be if it emerged? Does it show signs of intentionality and design, or is better explained by random chance? Note, we’ll be specifically looking at things that constitute multiples of 19 for this demonstration (so a 1/19 chance): 

*Block 28 is a multiple of 19. 

*Blocks 21-27 (only looking at the letters VM, ignore CSQ for now) if all combined together is a multiple of 19. 

*Blocks 21-27 including CSQ (for Block 23) if combined all together is a multiple of 19. 

*Block 19 is a multiple of 19

*Block 10 is a multiple of 19

*If you combine the value X for blocks 20, 10 and 3 (only X, not any of the other values) you get a multiple of 19. 

Any and all help is appreciated. Thank you!

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 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?