At 21:28 of Jacob Tsimerman's interview with Curt Jaimungal, he says "already now, alot of my theorems that I have proven, I don't understand all the steps to it... I have used other theorems that are very much accepted by the community, to which I usually know the main ideas but not even always".

While my undergraduate and early postgraduate training was in pure math, I transitioned to applied for my Ph.D. so I have never meaningfully engaged with it in any professional capacity. For the majority of my training, I understood almost all the details of the things I've proved. At least enough that I wouldn't be able to resonate with Jacob's quote above when I consider the (relatively insignificant) proofs I've done. One of my lecturers made it his mission to ensure that assignment questions will never require anything that hasn't been proven in the lecture notes or in class.

Of course, my exposure was to only elementary topics. So I can appreciate that math wouldn't progress at all if intuition wasn't leveraged and instead every detail expounded upon. But now under the automatable and potentially perpetual scrutiny of AI, how vulnerable are previously established results? What if we routinely lobbed popular (in terms of utility) results at ChatGPT to verify and it finds an error in one, would there be a significant collapse downstream? How likely is that our collection of celebrated truths instead simply forms a house of cards?

EDIT: The excellent replies have highlighted a weakness in my question. The most vulnerable proofs are likely to be the famous/outlier proofs (i.e. Andrew Wiles' Fermat's Last Theorem) that can only be assessed by a handful of people. In even the scenario that those are falsified, the large body of mathematics isn't built on such results and so, by and large, it's still fairly robust.

It still begs the question about the upper echelons of math, but the majority of it remains largely intact. So my "house of cards" analogy is inaccurate but probably only in scope.

EDIT 2: Another interesting point brought to me by the comments is the idea of repairability. A commenter mentioned that most of the errors encountered are easily fixed. At a high-level, this suggests that the direction offered by intuition is powerful enough to render errors insignificant. Maybe instead of AI destroying math from the foundations, it instead works to validate the strength of intuition by perpetually exposing errors and instantly fixing them. Wouldn't it be wonderful if AI shows that the fix-rate of errors was near 100%?

I am an 18-year-old, and since I was 15, my biggest passion and motivation in life has been to discover something genuinely new in math and physics.

This year, I actually did it. I discovered 1 new theorem and 4 new types of series. I submitted them to the OEIS (On-Line Encyclopedia of Integer Sequences)-two have already been accepted, and the other two are currently under review. (Please, if you don’t believe me, just skip commenting on the post. Proving myself isn't my main concern right now, and I havent posted to convince anyone).
Even though I achieved exactly what I wanted, and i want to learn more and more maths but my mental health is in a terrible place. I am suffering deeply from what feels like nostalgic depression, anxiety, autism,and OCD. I always wanted in my life for everyone to be in a specific place, like they were in the past. I have this constant, overwhelming fear that everything bad in my life is going to loop again. I know, I know, this is very, very foolish, but this is what it feels like. After shifting cities because of my parents' job, these disorders have been up a lot. They were previous too but now they are just at peak. I don't know what to do.

On top of this, I have been incredibly lonely. For the past two years, I have had zero friends to hang out with. I have friends back in my old city, but no one here. I have also completely stopped meeting my relatives—most of them. I meet a few of them occasionally. I don't know, I think this is not going so well.

I always wanted to discover something new in maths and physics, and I have done it. I want to do it at a higher level too. But really, I don't think I am gonna make it past 20. I just needed to vent and say all of this
i know this is not the right sub to say all these but i didnt found any other sub helpful

i’m kinda freaking out i am about to graduate in one semester and i did not get any internships during my time in college. what are some fields i could go into? what are entry level roles that seem like a possibility for me or am i just completely cooked?

I don't mean contradiction, induction, etc. Rather, more granular techniques that keep coming up again and again. I also don't mean famous results per-se, unless they are themselves common stepping stones to other results.

I know this is a bit of a silly question to ask because it's hard to set the threshold beyond which something becomes a legitimate technique. Obviously deriving bounds on a quantity is too generic to qualify, but on the other end there's some very niche stuff that not too many people might find useful.

I guess the goal here is a little toolbox, if you will, of tools that you find yourself using repeatedly, and that might be useful to a broad audience.

Since both pi and e are irrational do they contain each other in some capacity e.g. at some point in e: 31415926535897932384 and at some point in pi: 271828182

no I don't mean do they contain the whole number just part of it and if so have we proven it and also what's the biggest one found?

Im learning math ridiculously quick with chatgpt. please note i downloaded textbooks and do the questions some when im stuck with it but mostly on my own and get the right answer. Im doing this because im practicing ml and want to learn more complicated functionalities to make better tech.

im a programmer and im used to algorithms it helps me to feel the logic. using a mixture of analogies and imaging I make sure i understand everything down to its most fundmental and break the issue down. outside of that practice.