Making, testing and proving conjectures

[trees and plants]

Hey there .After making a conjecture whether in math or physics how do I check if it is on the right way?By applying special cases?About making the conjecture?Just connecting objects and forming relations?The proving part is a bit difficult but what about it?Is it also about a bit of doing it in luck, not completely purposely?When trying to prove because I am still a student, how do I do it?Generally guessing and checking, but something will lead me to the guesses I am doing?Perhaps having open books and searching for what to answer?Will that help sometimes?I am trying to learn things in math and physics but I mostly get stuck in many exercises.How to solve this?Thank you.

 

[mfb]

Ideally you find a proof, then it's not a conjecture any more.
If you can't find a proof, look for counterexamples as much as you can (this can also help finding a proof, by understanding what makes counterexamples impossible). If you find a counterexample the conjecture is dead (but might work with additional restrictions). If you fail to find them despite searching a lot then it might be an interesting conjecture. As an example, if you think all integers have a given property, then showing that the first few billion have that property is a good start - but not a proof, of course. You might also discover a proof that the conjecture is true for some special cases.

How to prove something depends on the thing you want to prove. There is no general approach "this is how you do it", otherwise we would have more proofs for long-standing problems.

 

[trees and plants]

Ideally you find a proof, then it's not a conjecture any more.
If you can't find a proof, look for counterexamples as much as you can (this can also help finding a proof, by understanding what makes counterexamples impossible). If you find a counterexample the conjecture is dead (but might work with additional restrictions). If you fail to find them despite searching a lot then it might be an interesting conjecture. As an example, if you think all integers have a given property, then showing that the first few billion have that property is a good start - but not a proof, of course. You might also discover a proof that the conjecture is true for some special cases.

How to prove something depends on the thing you want to prove. There is no general approach "this is how you do it", otherwise we would have more proofs for long-standing problems.

Is there a general process that scientific researchers follow to reach a probably correct conjecture?Not proved conjecture but probably correct statement for a theorem. Another question I have is a person having proved a conjecture or a question, will the proved result and the proof be accepted or is it necessary to be something interesting? I have problems with interest. Thank you.

 

[mfb]

Is there a general process that scientific researchers follow to reach a probably correct conjecture?

It's trivial to find an infinite set of correct statements. But almost all of them will be boring. "2 is larger than 1", "3 is larger than 2", ... - these are all trivially true, of course, you can find a bit more convoluted examples that will need a bit more thought.
If there would be a general process to produce interesting new results these results wouldn't be interesting.

Another question I have is a person having proved a conjecture or a question, will the proved result and the proof be accepted or is it necessary to be something interesting?

Define "accepted".
If you prove something that's trivial no one will care about it, yes.

You ask really vague questions but you seem to have something more specific in mind. It would help to be more specific.

 

[trees and plants]

It's trivial to find an infinite set of correct statements. But almost all of them will be boring. "2 is larger than 1", "3 is larger than 2", ... - these are all trivially true, of course, you can find a bit more convoluted examples that will need a bit more thought.
If there would be a general process to produce interesting new results these results wouldn't be interesting.Define "accepted".
If you prove something that's trivial no one will care about it, yes.

You ask really vague questions but you seem to have something more specific in mind. It would help to be more specific.

Generally I read mostly about my courses at my math school at university like in algebraic structures topics like generation of groups, subgroups, morphisms between groups other times I read about topology of metric spaces specifically connected metric spaces, complete metric spaces they provide theorems but I want different results as theorems.These are interesting and their proofs but I think more progress probably has been made. I am interested in way too many mathematical topics perhaps all of them and I do not know which of them to prefer most or choose mostly, it is difficult for me. What exactly is trivial?Like obvious?Something way too easy or not very interesting like 1+1=2? If I try to prove questions or conjectures in topics like minimal submanifolds in hermitian geometry or not so modern topics like linear algebra or multilinear algebra, tensor analysis, random graphs, theory of polynomials will people care about what I proved?Thank you.

 

[jedishrfu]

Well it has to be an interesting conjecture. A conjecture is a suspicion that someone discovers that seems true in test case after testcase and can’t be readily proved false.

Some example would be the Collatz conjecture or the twin primes conjecture.

https://en.wikipedia.org/wiki/Collatz_conjecture

https://en.wikipedia.org/wiki/Twin_prime

The famed mathematician Paul Erdos often said that there are some conjectures that we are just not yet ready to prove.

Here’s a few conjectures named in his honor or because of his work:

https://en.wikipedia.org/wiki/List_of_things_named_after_Paul_Erdős

 

[mfb]

If I try to prove questions or conjectures in topics like minimal submanifolds in hermitian geometry or not so modern topics like linear algebra or multilinear algebra, tensor analysis, random graphs, theory of polynomials will people care about what I proved?

If you find a proof for something that's generally known as conjecture then this is certainly worth a publication.
If you find a proof for a question you found yourself there is a good chance someone else found a proof before - make sure to check the literature.
Generally: If in doubt, ask someone more experienced in the field.

 

[trees and plants]

If you find a proof for something that's generally known as conjecture then this is certainly worth a publication.
If you find a proof for a question you found yourself there is a good chance someone else found a proof before - make sure to check the literature.
Generally: If in doubt, ask someone more experienced in the field.

What about making my own questions and trying to answer them rigorously?I have tried it but it seems to be very difficult I do not know how to answer the questions.How do scientific researchers do this? Someone could ask anything probably on a topic or topics by connecting objects and forming relations.For example in group theory by making questions connecting subgroups with morphisms between those subgroups and generation of groups will that provide a questions that with its answer will be interesting?Will ever an answer to a question with its question if it till then remained unanswered be considered not good enough or rejected?What are the requirements for a question with its answer be considered accepted for publication or not for publication?What I mean is if I make the efforts to answer those questions and take the time to learn and try to answer them eventually will my efforts be fruitful?

 

[mfb]

If something is easy it's not new (with very rare exceptions).

How do scientific researchers do this?

See previous posts.

Will ever an answer to a question with its question if it till then remained unanswered be considered not good enough or rejected?

I don't understand the grammar of that sentence.

What are the requirements for a question with its answer be considered accepted for publication or not for publication?

Be of sufficient interest to readers of the journal. Read the journal to see what is of sufficient interest its readers.

What I mean is if I make the efforts to answer those questions and take the time to learn and try to answer them eventually will my efforts be fruitful?

That depends on you.

 

[PeroK]

What about making my own questions and trying to answer them rigorously?I have tried it but it seems to be very difficult I do not know how to answer the questions.How do scientific researchers do this? Someone could ask anything probably on a topic or topics by connecting objects and forming relations.For example in group theory by making questions connecting subgroups with morphisms between those subgroups and generation of groups will that provide a questions that with its answer will be interesting?Will ever an answer to a question with its question if it till then remained unanswered be considered not good enough or rejected?What are the requirements for a question with its answer be considered accepted for publication or not for publication?What I mean is if I make the efforts to answer those questions and take the time to learn and try to answer them eventually will my efforts be fruitful?

Why don't you try the November Maths Challenge:

https://www.physicsforums.com/threads/math-challenge-november-2020.995557/

There are several unanswered questions there.