yeah, that's what i was guessing. i think it just has more interesting potential consequences than a lot of other conjectures.
e.g. the twin prime conjecture is interesting and any methods used to settle the matter would probably represent a huge advance in understanding, but i don't know of relative results where people assume its truth and deduce other things. maybe there are some.
Twins, I found a note about it, and by chance I found a question here on the site about me on the same note, and the answer was that this is normal happens:)
@M17 check out my recent posts. There are twin prime counting solutions in an interval
@M17 that's a nice pattern :) I have found that such things will stop occuring somewhere in the hundreds
You have to write a script to test small cases, and you most likely will run into a counterexample. If you do not, then you should post a conjecture to MSE to timestamp it
There is someone who answered me, someone I found gave the same idea as me and replied that it doesn't stop, which is computationally normal. Unfortunately I did not find the question.
i don't think ted was suggesting that anybody be screwed over. it's a fact that famous problems amenable to arithmetic investigation have seen a lot of investigation.
Sometimes it's better to heavily apply your analytic brain to a formula, disregarding hidden algebraic structure, just because it's easier than generalizing everything small thing
I think there are border cases of elementary vs advanced. If most people have to spend hours analyzing an idea because it appeals to some isomorphism such as CRT in abstract algebra, then it's borderline advanced, but somewhat elementary as well.
@TedShifrin I apologize, I'm just disturbed by the treatment of elementary approaches, the downvotes, negative oppinions etc. There will be elementary proofs that will rival the advanced approaches. The smallest proof wins. Elementary math can be elegantly written
Take Erdos, he did things with elementary math and published thousands of papers. He took the elementary and took it to an advanced level, so that it was no longer elementary.
There will be an elementary proof of Twin primes.
Not by everyone's standards, but by graduate level standards yes
elementary and experimental math is great, but i do think that the threshold for getting too involved in a conjectured pattern relating to prime numbers should involve checking more than, say, two-and-three digit examples.
you don't even need factoring or a decent primality test. you can download a large list of prime numbers and search it like a lookup table for patterns.
If there were an elementary proof, I imagine that it would have already been published. Elementary tools seems to be insufficient, which is why more advanced tools are created.
There are elementary patterns we just can't see them. Parameterized hyperplanes causing periodic values etc. It just takes a computer to search for them, but otoh the number of formula forms you have to check is infinite, so it takes creativity to know what to look for
i wouldn't rule out an elementary proof of FLT, but i expect that if it one is found it will put a lot of pressure on what people mean when they say 'elementary'.
You also need to know how to code, the computer will help you check out certain ideas more efficently
I recommend Python / Sympy
When you need fast computation 1000x faster than Python because something is just hard to compute by nature, you code it in C++
When things get advanced though like in diagram chasing, there don't exist good software yet for that
When we have those advanced tools (Lean4 has GUI widgets already), it's gonna be crazy.
People will be cheating even in grad school, but I don't consider it so.
It's just using a calculator...
Once you've done 20 diagram chases by hand, you've earned the right to use a diagram chaser app
When you've proved the Snake Lemma 3x because you keep forgetting it, well a calculator with nice visuals will help you remember once you start using the tool well
If you get Python 3.x installed on your system (hopefully Windows), you can borrow my IDE's license key for WingWare and I can show you how to code a 50-100 line test for your idea.
@M17
If you have a goal for coding, that will help you learn, and you have one
When you go by tutorials that will just get too boring
Use tutorials (a piece of them) as a reference
Google is king because without it you wouldn't be able to search and code and copy / paste snippets as fast
Remember, the code won't prove anything, but just verify it for the first few 10 - 1000+ examples, which you cannot achieve by hand calculation!
Otoh, Lean4 will help you prove things! And you could publish that formal proof (it's a proof assistant language). It's more difficult than C++ and I don't know it
@M17 we have to Zoom meet or use TeamViewer to do a tutor session for Python
But I can't right this moment, but later tonight Pacific time around 7:00pm and later I can
Google "Python 3 installer download".
Look at page, learn how to check your OS for 32-bit or 64-bitness, download the right installer
Once you have that installed, we can get to the tutoring part much quicker.
When you run the installer this is important, remember to click on any "custom install" button and install to something like C:\Python39 in case of Python 3.9 for example, and also "Add Python to Path"!! The last one is most important, but you should check everything including install debug symbols.
If you'd like to chat further, you have to click my avatar and go to "Start a new room with this user".
$(x\cdot y) + 26 = $ prime, Hover over or right-click that math text I made, and go to "View as MathJax" somewhere, and then you can copy / paste that into your chat edit box. Alternatively, you just surround your formula with $ or $$
I.e. single dollar sign or double dollar sign $$
The second one is for block display in a new paragraph
Hi guys, how are you? Can someone give me some sources to get challenge exercises? (To do a parallel here, its the same that LeetCode is to science computers).
there are sequences and series tags on math.SE also. might be worth browsing them although the bias might be in favor of problems more difficult than those that might appear in a calculus classroom.
I'm not supporting Russia or anything in light or recent events. I just have used that website before
Idk where the author is from or where it's hosted though
Why are they going to war, this is stupid. What a sham, pyramid scheme, sending boys to fight dirty wars for secret selfish reasons
If everyone studied math, we would be killing each other on the battlefield shouting that one theorem is better than another theorem. Right? No, I think we're a peaceful bunch
If the people or Russia would just open their hearts and study their forbidden math book repository, the world would be a safer place.
Well, more like 15, and I'm gonna sell some like Hartshorne which I'm not ready for yet
I can work a few exercises but not like a pro
For instance I did a diagram chase proof of a sheaf exercise and the solution manual on Github has it done with two sentences and a reference to abelian category axioms
I think the most non-trivial part of a diagram chaser app (hypothetical) would be graph layout and information hiding
The logic / type theory part of it is well-studied
But difficult to compete with such a system as Coq, Isabelle, or Lean4
even professional mathematicians tend to give proofs for their theorems that are not the most elegant ones out of all possible ones. A lot of proofs of major results are simplified and refined later, or sometimes completely different proofs are found
@PenAndPaperMathematics Eisenbud is good, but it is sometimes too wordy for my taste. Matsumura is more concise
I know, and that's highly likely the more complex the proof gets
I did Matsumura a few chapters. I might go back to it, but I think that's one I will need to purchase because the digital typesetting is wrong in my copy
@LukasHeger And the homological algebra in a CA book is astounding, that's what I'm really interested in. I want to be able work with it for my own elementary attacks, but which naturally tend toward homology/cohomology once you've found a way to encode the problem nicely
Eisenbud has lengthy motivations and intutions, which are great, but they also mean that you have to read more text to cover the same material; Matsumura is concise and to the point, he manages to cover a lot of material in under 300 pages, whereas Eisenbud is a massive tome. So both have their merits
@XanderHenderson True. "Dr. Who and the Daleks" was not really in keeping with the Doctor Who series. They tried to reconcile them, but didn't do that well.
xander: retroactive amendment of my statement to "it's like in that show entitled Doctor Who, which is often and erroneously referred to as Dr. Who, and is the title of a show and not a character. they recast the character every few years."
@robjohn A friendly reminder to help you not miss another occasion for "dressing up" (I'm still mourning the absence of the mean heart for St. Valentine's day!) In the next week you'll want to dry clean your attire so you're good to go, for our next "holiday" (which is before EASTER!!! Hint, hint. :-)
@XanderHenderson I have gravatars for Valentine's Day, St Patrick's Day, Easter, Memorial Day and Independence Day (US), Halloween, Thanksgiving (US), and Christmas. None for Purim, yet.
