The shortest papers ever published (2016)
paperpile.com
paperpile.com
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 still generate new prime numbers with this technique.
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.
Phil Plait used to complain about people who told him the moon wasn't visible during the day.
https://dosequis.colorado.edu/Courses/MethodsLogic/papers/Wa...
"Can a good philosophical contribution be made just by asking a question?" https://doi.org/10.1111/meta.12599
The guy who published it seems kooky as well. Would love to interview him some day!
There's some solid microbiology here which underpins the quality of the science.
“?”
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.
"Molekularstrahlenablenkungsmethode"
The journal turned it down.
https://www.aip.org/history-programs/niels-bohr-library/oral...
https://www.sciencedirect.com/science/article/pii/0925772195...
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.
> We have posed a fine (in our opinion) open problem and reported two distinct “behold-style” proofs of our advance on this problem.
There's also a linked PDF which, if I'm reading it correctly, trivially proves n²+1 is impossible.
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.
Also, they're permitting overlapping of triangles, right?
If that's the case, why can't you just add an arbitrary + 1 wherever you please and call it a day?
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.
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.
This leaves the question: is n^2+1 possible?
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?
Sorry, I'm slow on these things.
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.
[0]: https://en.wikipedia.org/wiki/Square_packing#Square_packing_...
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.
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.
The second figure is actually showing another arrangement of n²+2 small unit equilateral triangles covering an equilateral triangle of side length n+ε.
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.
Fiengo, Robert, and Howard Lasnik. 1972. “On Nonrecoverable Deletion in Syntax.” Linguistic Inquiry 3 (4): 528. https://i.imgur.com/vLntfCp.jpg
The Shortest Papers Ever Published (2016) - https://news.ycombinator.com/item?id=15737611 - Nov 2017 (93 comments)
1: https://www.sciencedirect.com/science/article/abs/pii/003191...
so... egg?
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.
May I ask if these "faux sciences" contradict your political or ideological positions, and is it possible that is the real issue here?