What is Conjecture: Definition and 227 Discussions
In mathematics, a conjecture is a conclusion or a proposition which is suspected to be true due to preliminary supporting evidence, but for which no proof or disproof has yet been found. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.
Any set of a series of numbers consisting of increasing integer members, all of which are determined by a common proposition or characteristic, will always be infinite in size.
Examples…
Prime numbers
Mersenne primes
Odd perfect numbers(if they exist)
Zeroes of the Zeta function
Regardless...
Hi there I am trying to get into topology
I am looking at the poincare conjecture
if a line cannot be included
as it has two fixed endpoints
by the same token
isn't a circle a line with two points? that has just be joined together
so by the same token the circle is not allowed?
Can i get a...
I did a study of the Collatz conjecture and found that all even numbers can be removed from the Collatz conjecture because even numbers act as the connecting links between odd numbers.
What do you think about it? Is it a breakthrough in 3n+1 problem?
27 -> 41 -> 31 -> 47 -> 71 -> 107 -> 161 ->...
The intense gravity near the event horizon causes complementary particles to pop into existence spontaneously. As local space-time is continuous through the EV, the same would be happening just inside the EV, only more so as the gravity field and gradient is greater. So near the singularity...
Note as soon as the term 3N+1 become divisible by a power of 2 we can repeatedly divide by 2.
For the proof below we rearrange the sequence so it becomes:
First step:
If N is odd, multiply by 3 and add 1.
Each next step:
- Repeatedly divide by 2, as many times as the number k, which is...
This has bothered me for quite some time. This drawing represents the plastic ring around your typical 6-pack of Dr. Peppers, Coke or most any 6 pack of something. The ring consists of 6 loops for the drinks, two small internal loops, and one handle loop. As any good dolphin lover would do...
The Toeplitz Conjecture (better known as the inscribed square problem) states that all Jordan curves have an inscribed square. It has been stated in the early 1900's and remains an open problem.
I drew a square and then making a ton of curves that touch its four vertices:
This shows that the...
Lets say you were trying to prove a math statement when you realize that you can use a conjecture (say, Goldbach's conjecture) to finish the proof. If you don't have the time or the brains to prove it, how many cases of Goldbach's conjecture do you prove so that you can use it in your proof?
Proof:
The proof is by induction.
(1) When ## n=4 ##, the statement is ## p_{4}<p_{1}+p_{2}+\dotsb +p_{3} ##,
which is true, because ## 7<10 ##.
(2) Now assume ## n=k+1 ##.
Then ## p_{k+1}<p_{1}+p_{2}+\dotsb +p_{k+1-1}\implies p_{k+1}<p_{1}+p_{2}+\dotsb +p_{k} ##.
Thus ##...
Proof:
Let ## n ## be an integer.
Then ## 2n=p_{1}+p_{2} ## for ## n\geq 2 ## where ## p_{1} ## and ## p_{2} ## are primes.
Suppose ## n=k-1 ## for ## k\geq 3 ##.
Then ## 2(k-1)=p_{1}+p_{2} ##
## 2k-2=p_{1}+p_{2} ##
## 2k=p_{1}+p_{2}+2 ##.
Thus ## 2k+1=p_{1}+p_{2}+3 ##...
Proof:
Suppose ## 5777=p+2a^2 ##, where ## p ## is either a prime or ## 1 ## and ## a\geq 0 ##.
Now we consider two cases.
Case #1: Suppose ## p ## is a prime and ## a\geq 0 ##.
Let ## p=2 ##.
Then ## 5775=2a^2 ##.
Thus ## a=\pm \sqrt{2887.5} ##,
which contradicts the fact that ## a\geq 0 ##...
WHat is swampland conjectures in string theory? I cannot find any online resources that can break it into laymen but as I watch Cumrun Vafa's presentations and its wikipedia description, they are string theories that are outside the landscape. So I am thinking that universes with Swampland...
I know that it has been proven that for the expression x^a -y^b = 1, only has this one integer solution, where x = 3, a =2, y =2, b = 3. I am interested in knowing if there is a proof for this expression: 2x^a - y^a =1 in which there are integer solutions for x,a, and y or if no integer...
Mihăilescu's theorem proves that Catalan's conjecture is true. That is for x^a - y^b = 1, the only possible solution in naturual numbers for this equation is x=3, a=2, y=2, b=3. What is not clear to me is this. Does Mihăilescu's theorem prove that the difference between any other two...
There is a fascinating story that I'm sure a lot of you have followed.
In 2012, a top mathematician, Shinichi Mochizuki[1], has claimed to have solved the ABC conjecture[2] (an important longstanding problem in number theory), using his own very unique, complex, and abstract Inter-universal...
The new paper "Testing a conjecture on the origin of the standard model" Eur. Phys. J. Plus 136, 79 (2021). https://doi.org/10.1140/epjp/s13360-020-01046-8 has been published. Springer allows to read it online at https://rdcu.be/cdwSI .
Over 100 numbered experimental predictions about physics...
Hello,
I have posted a problem (on math stack exchange) I was given for fun by an uncle, who doesn't know whether the proof is possible to establish. I tried my best and failed so far. I don't think I can solve the problem with my current knowledge and I would love to know if you can find a...
The Collatz problem is perhaps the only unsolved math problem I actually understand. It "feels" like a proof would be trivial, though obviously it isn't. Been playing with different variations in hopes of understanding it better. Is it a set problem (proving there's no intersection between two...
I am working on a presentation for a course in general relativity and my topic is the stability of black holes. In many of the references and articles that I have found, the author asserts the importance of the conjecture but offers no reason. So I ask: Why is the black hole stability conjecture...
The swampland conjecture https://arxiv.org/abs/hep-th/0509212 is currently a very hot topic. Can someone explain, in simple terms, what exactly the swampland idea is? In particular, the conjecture states that the string landscape is surrounded by an even larger swampland of consistent-looking...
You have an infinite supply of square sheets of paper. You are going to secure these sheets on an infinitely large bulletin board by using thumbtacks. You must secure all four corners of each sheet however you may slightly overlap the sheets so that one thumbtack could secure up to four sheets...
When Grisha Perelman submitted his proof of the Poincare conjecture, he may have been reasonably sure that it contained no mistakes. But he could not have been 100% sure as he is, after all, human. Each time it was checked, say by the referee of an academic journal, the probability that it...
I read somewhere that this was proved sometime in the 80's, but that same source didn't mention that the proof was wrong. Of course I would cite the source but I can't find it again.. Does anyone know of any specific reason why this is still a conjecture?
I realize that for these sorts of...
I have been interested in the attempts to prove this Conjecture since 2000 and like many others (eg Ken Conrow) I have tried to find a convincing solution. Today I read on this forum what looks like a proof that there cannot be an internal cycle beyond 4:2:1 but I don't think the author realizes...
For instance Alain Connes has dedicated work to Riemann's Hypothesis, who would fit the analog for this on Hodge's Conjecture? Has there been any recent progress done in the field? Since it's quite an esoteric subject of matter and with work on it being done at the best gradually to my knowledge...
“If the weak gravity conjecture is right, loop quantum gravity is definitely wrong,” said Nima Arkani-Hamed, a professor at the Institute for Advanced Study who co-discovered the weak gravity conjecture.
source...
Correct me if I am wrong, but my basic understanding of how the Chronology Protection Conjecture (CPC) would work is that, as virtual particles created from the quantum fields of the vacuum would traverse a wormhole and arrive in the past, they would then travel back into the wormhole alongside...
The essence of time is change. A driver of change is the 2d Law of Thermodynamics. The expansion of the universe facilitates entropy. These relationships keep me wondering if they are a symmetry. Comments please.
Dear All Gravitinos,
I write this post here to discuss a new conjecture on resolutions of the schwarzschild singularity and the physics interpretation for the micro states of black-holes (arxiv: 1606.06178, published in Nucl. Phys. B2017,02,005...
Hello, everyone. I've been trying for quite some time to figure out what's up with the paradigm of quantum interest conjecture and the possibility to generate negative energy (even only theoretically speaking). Only from pure interest I have read a bunch of papers on the matter, however I feel...
With the lonely runner conjecture, can the runners run along a circular track of any diameter or does the conjecture require that they run along a unit circle?
There is a graph showing n on its x-axis and its total stopping time on its y axis.
From here we can see that the points on the graph are not random at all; they have some kind of geometric pattern that is due to the 3x+1 in the odd case and x/2 in the even case. I have seen many attempts to...
Here is the Wikipedia article on the lonely runner conjecture
https://en.m.wikipedia.org/wiki/Lonely_runner_conjecture#
I have some inquires about it.
Firstly I am right in thinking that "pair wise distinct" means that the speed of all runners are different?
Also does every runner have to be...
I was examining the AKS and discovered this conjecture.
Please prove the following true or false.
Let n be an odd integer >2
then n is prime IFF
$\left(
\begin{array}{c}
n-1 \\
\frac{n-1}{2} \\
\end{array}
\right)
\text{ $\equiv $ }
\pm 1$ mod n
Hello all,
I recently starting studying graph theory in my free time and have become very interested in the Reconstruction Conjecture. Since I am new on the subject I am not sure where to start my search for additional information/insights/papers on the topic, I thought I would ask here for...
Now, first off I am punching well above my weight here but oh well. I am doing an extended project on theoretical methods of space travel and was doing some brief reading in the middle of writing about Miguel Alcubierre's warp drive for hyper fast travel within general relativity. I came across...
Here is an article about a conference discussing Shirichi Mochizuki's claimed proof of the ABC Conjecture.
http://www.nature.com/news/biggest-mystery-in-mathematics-in-limbo-after-cryptic-meeting-1.19035
So, I was reading the Wikipedia page for Legendre's Conjecture and I came across this:
"Two stronger conjectures, Andrica's conjecture and Oppermann's conjecture"
My question is this: What does it mean for one conjecture to be 'stronger' than another?
The following popped into my head and I am curious whether it is already a known relationship and whether it has an utility in math/physics. It is a follows: Where x and y are consecutive, whole numbers, the following is true: x + y = y2 - x2
Does the Poincaré's conjecture for a three dimensional manifold involves only simple connectedness or is it meant that the first and second homotopy groups are trivial ?
Since in the first case the conjecture seems to me wrong whereas in the second true.
Thanks.
I was fooling around with the Prime Zeta Function just recently.
Prime Zeta Function, P(s), is defined as Σ1/(p^s), where p is each successive prime. When inputting various positive integer values for (s) on wolfram alpha, I noticed a pattern. Well, an approximate pattern, I should say.
My...