61–70 of 84 posts
1: https://www.sciencedirect.com/science/article/abs/pii/003191...
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.
I’m sure the faculty were thrilled with this tradition. Less pages for them to read!
Not a paper, but 2×3×5×7×11×13 + 1 = 59×509 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 sti…
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.
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.
Earlier quoted context omitted.
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.
Not a paper, but 2×3×5×7×11×13 + 1 = 59×509 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 sti…
https://www.sciencedirect.com/science/article/pii/0925772195...
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: “?” Publisher replied: “!”
It eventually leads me to hear the MGS exclamation mark sound when I play or assist to a game of chess and that I spot a good move.
Earlier quoted context omitted.
I was curious if GPT-4 Vision could explain it: 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…
It couldn't.
This neglects the classic paper published in The Annals of Improbable Research, The Effects of Peanut Butter on the Rotation of the Earth: https://improbable.com/airchives/classical/articles/peanut_b...
Can someone explain the n^2 + 2 triangles 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.