I call for the paper to be retracted! Peanut cultivation requires water with the corresponding extraction of groundwater. Extraction of large amounts of groundwater can cause land subsidence and change the local topography. If done on an extremely large scale...There could be a tiny effect on Earth's rotation.
is a short counter example to the widespread misconception that adding one to the product of the first n consecutive prime numbers always yields a prime number.
The reason you get away with this in the infinitely-many-prime-numbers proof is that the new number may not be prime, but can be written as a product of primes that are distinct from the first n primes. Thus you still generate new prime numbers with this technique.
I think this is not a problem because 59 nor 509 are in the list of prime numbers used (on the left side). Euclid's proof merely states that for every list of prime numbers, there's a new one, not in the list. https://en.wikipedia.org/wiki/Euclid%27s_theorem#Euclid's_pr...
Yes, that's the point -- Euclid's proof doesn't require that, it works like you said; but it's a widespread misconception that it does work that way. It's a counterexample to the misconception, not to the real proof.
Under the assumptions of the contradicting statement, 59x509 is prime. You can’t factorize a number iff there are no prime divisors, and the right hand definition for prime numbers is much simpler for the purposes of the proof; no extra “prime” generation necessary.
You are right, but I wouldn't emphasize this in the proof, for the following reasons:
1. Since you assume a contradictory statement, you can actually derive everything you want. So saying "it's true in that context" is pretty meaningless.
2. I think it adds an unnecessary step in the proof. "This new number is not divisible by any prime, therefore it is prime, contradiction as it is not in the list" compared to "It is not divisible by any prime, contradiction since any number is divisible by a prime". I think that is confusing.
3. For didactical reasons. It can leave the reader/student with the wrong impression that multiplying the first n primes and adding one always creates a new prime.
When I saw this paper about “Royalactin” in Nature [1], it was fascinating subject matter for one but I was impressed that it was a single author non theoretical original research article in a high impact journal. I thought it was baller. Have been trying to search for anything similar published in recent times and have come across empty. I feel it’s emblematic how impossible it is to make any great biology breakthrough nowadays as a lone wolf. But one can aspire!
The guy who published it seems kooky as well. Would love to interview him some day!
I don't see how LLM would allow someone to do all of the experiments found in this paper. How do you think an LLM would promote more lone wolf science?
There's some solid microbiology here which underpins the quality of the science.
Vaguely related but I remember the tale of an author on vacation after publication of his latest novel. He telegramed his publisher to enquire about reception:
Which is weird because it's a perfectly simple and understandable word. Did they want to rephrase it to the more complex "Methode zur Ablenkung molekularer Strahlen"?
In computational geometry, Raimund Seidel wrote a paper that proves an upper bound theorem for polytopes in two sentences in the abstract. The rest of the paper just comments on the result.
My high school math teacher told as about how students at his university competed for the shortest bachelor paper.
Not getting a paper published is par for the course, but having to retake your bachelor examination is quite the hassle. The risk and associated bragging rights seemed quite big.
It exhibits two distinct constructions both of which demonstrate that n^2 + 2 unit equilateral triangles are sufficient to cover an equilateral triangle of side n + ε. The obvious area argument shows that at least n^2 + 1 are required.
A small modification of the second figure can show that for any non-equilateral triangle, n^2 + 1 such triangles will cover a similar triangle of length ration 1 : n + ε; it remains (as of 2010, at least; see [1]) an open problem whether a construction of n^2 + 1 triangles exists in the equilateral case.
ε is used to denote an arbitrary small value, which isn't zero. Overlap is indeed required here.
Let's say you have a large equilateral triangle of side n. Covering it with triangles of side 1 is pretty easy: you build a pyramid out of them without any overlap. That requires n^2 smaller triangles. Now let's say you make the large triangle sliiightly larger, so it'll have sides of n+ε instead of n - for example we gone from 11.0 to 11.00001. How many smaller triangles do you need to cover it?
Obviously n^2 isn't going to be enough - because that was exactly enough to cover a large triangle of side n. Our slighty-bigger triangle is slightly bigger, so it has a larger area. We're going to need at least one additional small triangle to cover the added area, leaving us with n^2+1 as an absolute lower bound. But just because it is a lower bound doesn't mean it is actually possible - you'd first have to demonstrate that it can actually be done.
This paper demonstrates two different methods of constructing it with n^2+2 triangles, providing an upper bound which is definitely possible. This means we still don't know the exact number of triangles required, but we do know it is definitely bigger than n^2 and definitely smaller than or equal to n^2+2.
Q1: So the second one is essentially "pushing things down" from the top as in the extra space is being accounted for by those 2 additional triangles?
Q2: The problem is non-trivial because it appears to open up a trapezoid somewhere in the stacked triangle solution that can't be covered by a single triangle?
Q3: This sounds provably impossible unless there's another way to cover the n triangle other than stacking. It sounds like the solution space is pretty finite and can be manually exhausted. Is there something I'm missing?
Q1: If you look at figure 1, you can see that the "down" triangle row is sticking out a bit to the left and to the right. This allows the "up" triangles to move down and to the side a little bit. Both the "up" and the "down row are one triangle bigger than then would've been in the non-ε variant, which allows the extra space being covered.
Q2: A trapezoid is left at the bottom if you just stack triangles, yes. Other approaches will probably result in one or more gaps of a different shape.
Q3: There's an infinite number of ways you can arrange the small triangles, so an exhaustive search isn't going to help you. The interesting part is that there is a proof of n^2+1 being possible for all non-equilateral triangles, so there is definitely a possibility of it also being possible for equilateral triangles.
As you already noticed, there might be approaches beyond stacking. Look up "square packing in a square"[0] for fun, you get some really ugly-looking non-obvious results out of that.
Don't worry about it, I know just enough to understand the problem - half of the linked PDF is also beyond me.
> There's an infinite number of ways you can arrange the small triangles
I don't understand how this is - you have to eventually have a consolidated gap for your extra triangle, everything else needs coverage. It demands a level of efficiency that confines the possibilities.
This isn't a packing problem as in gaps are permitted, it's a coverage problem as in, gaps are not.
You can overlap things at leisure but you quickly enter the efficiency problem again. Once your aggregate overlap is the area of one of your smaller triangles, it's no longer possible.
So as far as I can see those are the bounds. You're allowed to overlap and extrude up to some function of the (area of the smaller triangle, n and the epsilon) and the gap that's created must be confined to fit inside the geometry of one of the smaller triangles.
It appears to be tightly bound enough to exclude exotic arrangements of the triangles.
Furthermore there's no novel arrangement possibilities you get once n becomes really really big because of the geometry confinement problem - so some exotic thing like dilating a row along an arc or skewing through some pattern isn't going to help you.
This means demonstrating for a very small n is sufficient. You've got the geometry of the trapezoid to bring to the confine of the gap while the sum of the overlapping and extrusions can't pass a certain threshold.
I bet it's within my ability to write a computer program to exhaust it and if I was better at the mathematical fancyspeak, there's probably an algebraic proof in here.
ε > 0. Yes, overlapping triangles. Add an arbitrary + 1 to what? You need to arrange the small triangles so that they cover the big triangle of side length n+ε. n² unit equilateral triangles cover a big equilateral triangle of side length n without overlap, so at least n²+1 are needed for side length n+ε, and the paper shows (not especially clearly, IMHO) that at most n²+2 are needed.
Figure 1 on its own is a pretty decent demonstration, once you zoom in quite a bit.
The annoying part about the paper is figure 2: it shows a different method of doing so, without mentioning that it is unrelated to figure 1. It is also drawn in a less obvious style, which really hurts its readability.
Yes, Figure 1 is okay, but Figure 2 is not very clear; most of the triangles are missing and have to be imagined. I also think the examples for a single n are not the best argument for the general case - the reader can extrapolate other diagrams and a proof but I think a slightly longer paper could be clearer.
I think the fact that you have to ask this proves that it is objectively a bad paper.
The whole point of academic papers is to contribute to the larger global knowledgebase. You acknowledge the work that was done before, you submit your contribution and then you suggest how people can build or expand upon your work. This paper in question is just trying to be a mic-drop, like a middle-finger to academia.
Generally research papers cover background context as their 1st and 2nd sections (at least IEEE format papers do). So normally a paper like this would start with an introduction section which explains what the paper is accomplishing and then the background section number two would explain context for where the author is coming from or what inspired research or background to justify its value. These sections would provide the context you are looking for and at the very least give references for you to go back and learn about it on your own. Even a few sentences would have been powerful here.
This paper does fail to really provide value in my opinion and is objectively a bad paper. With some additional context from the introduction and background this could be much more valuable. Less critical, but also important is to acknowledge limitations and suggest future research.
Now with all that being said, I'm not saying research papers are perfect. It is easy to find examples that go too far the other way, with far too much verbosity and pomp and circumstance. So I do at least acknowledge the statement being made with this paper that maybe all you need is two words. The reality is we should be somewhere in the middle. I read 3-10 academic papers per week, and the average page length is usually around 10 pages and really should be closer to 3-4. So i acknowledge the statement being made here, but this paper is clearly a protest, and not actually a productive example.
The paper presents a geometric problem centered on equilateral triangles. The key question is whether it's possible to use \( n^2 + 1 \) small equilateral triangles (each with side length of one unit) to cover a larger equilateral triangle that has a side length just slightly more than \( n \) (specifically, \( n + ε \), where \( ε \) is a small positive value).
The two figures provided illustrate possible arrangements of the smaller triangles within the larger one:
1. *Figure 1*: This demonstrates that \( n^2 + 2 \) small triangles can cover an equilateral triangle whose side is \( 1 + ε \). It's evident that the small triangles fit neatly inside the larger triangle.
2. *Figure 2*: This shows a different configuration where the large triangle has a side length of \( 1 - ε \). It seems to suggest that with just one fewer triangle (i.e., \( n^2 \)), the tiling is not possible for a triangle of side length \( 1 + ε \), but it may be for \( 1 - ε \).
The paper, although succinct, poses an intriguing tiling problem in geometry. The authors likely aim to stimulate thought and discussion on this particular geometric configuration and challenge readers to consider the conditions under which such tiling is feasible. Given the brevity, the paper might be a problem statement or a brief note, rather than a full research paper with exhaustive proofs.
Pretty damn close though! I haven't seen an explanation of what the second figure is trying to show so I'm not sure about that one. (And also their assertion that no further explanation is necessary is clearly bullshit.)
Its attempts at explaining both figures are totally wrong. Wrong side lengths, wrong assertion that the small triangles fit inside the large triangles, wrong relationship between the figures, complete misunderstanding of the second figure.
The second figure is actually showing another arrangement of n²+2 small unit equilateral triangles covering an equilateral triangle of side length n+ε.
Obliquely related, the shortest paper title I am aware of: H=W. By Norman G. Meyers and James Serrin, Proceedings of the National Academy of Science 51 (1964), 1055–1056.
Short abstracts such as ‘yes’, ‘no’, &c of papers whose titles are questions are amusing but not as helpful as those whose titles straightforwardly state the answer (e.g., ‘X is Y’ is preferable to ‘Is X Y?’ with abstract ‘yes’).
Brian Josephsons' paper on the superconducting tunneling effect that's now named after him was one of the first (and last) papers he ever published and netted him a Nobel prize. It's just 2 pages long I think [1]. Can't be sure as I cannot get the full text since it's paywalled, pretty sure it's only 2-3 pages though.
Hmm. I'm wondering how the short lengths of these papers might cause them to have lower information-theoretic entropy than the (in)famous "chicken" talk[1]?
What I do have are a very particular set of skills. Skills I have acquired over a very long career. Skills that make me a nightmare for people like you.
As a rule, when trying to convey information, I try to write and speak plainly. I try to avoid jargon.
I have found many academic papers in the faux sciences to be extremely dense and full of terms that are only known to the priests of that arcane subject (still subsidized by taxes as if the result is a common good).
If you have a point, say it. There is no need to write in legalese. When I see supposed research written like this, I assume it’s a grift written just for the tiny group of academics tenured in that subject, who review each others’ papers every year, buy each others’ books, and keep the perpetual motion machine of funding running until they hit retirement.
It is a valuable skill to be able to communicate in plain language, and a valuable skill to be able to understand dense language. Communication requires investment from both parties. If you can't be bothered to look up their terminology, that's all well and good, it's you're time to spend, but your lack of investment doesn't imply that it's a grift.
May I ask if these "faux sciences" contradict your political or ideological positions, and is it possible that is the real issue here?
I was hoping that LLMs would help with "writer's block", but their limited context window has been a PITA. Has anyone have had any success with ROPE scaled models or claude 100k?
Have you tried MemGPT or RAG in general? I'm not convinced long context alone will solve the issue (at least not very soon), some kind of retrieval and especially self-retrieval is good.
