 As we end lecture five, I'm continuing the conversation we've had over the last couple of lectures in our last video. In lecture three, we talked about theorems and the different type of theorems you have out there, like corollaries and limas and such. In lecture four, we're going to introduce the axiomatic method about axioms, definitions, examples, theorems, proofs, etc. And so I want to kind of continue in this conversation and talk about the idea of a conjecture. Remember, a theorem is a true statement that is true by a proof as opposed to an axiom, which is a true statement without a proof. That is, we assume it to be true for theoretical reasons. A conjecture is a statement, so it's related to axioms and theorems here. It's a statement, but we don't know whether it's true or not, but we believe it to be true. So we think it's true, but we don't have a proof yet, and we can't necessarily just axiomatically take it as a fundamental truth. It's something we believe to be true, but it's not fundamental enough to be the bedrock of our theory. And so let me give you an example of such. This is a very famous conjecture, at least it's a conjecture at the recording of this video. If someone proves this in the future, that would be fantastic, but at the moment, it's a conjecture. It's known as Golbeck's conjecture, which states the following, which you can see here on the screen. For every even integer n greater than 2, there exist prime numbers p and q such that n is the sum of those two prime numbers. And I want you to try to investigate that for a second. You take, for example, 4. 4 is 2 plus 2. 2 is a prime number. The conjecture does not require the primes to be different. Take 6. 6 is 3 plus 3. 3 is a prime number as well. Take 8, which is 3 plus 5. Take 10, which is 3 plus 7. Take 12, which is 5 plus 7. We could do what's next, 14. 14 would be 3 plus 11. And it could actually be more than one way to write it as a sum of prime numbers. We won't worry about that right now. Take 16. 16 is 3 plus 13. 18. 18 is 5 plus 13. 20. 20, we could do that as 3 plus 17. 22 would be 3 plus 19. We could do 24. 24, you could do as 5 plus 19. 26. Let's see, you could take 3 plus 23. And 28. You could do 5 plus 23. 30. You could do a 7 plus 23. And we keep on going. I'm going to stop here. I think that's plenty of examples to make a point. At least for the first couple, even integers, it does seem to work. There always seem to be at least two primes that come together and make it work. Now for larger even numbers, we could continue this in this manner. Just experimenting and see if we can find it. And we'll continue to find primes that satisfy this conjecture. In particular, in particular here, this conjecture holds for all even numbers up to at least 4 times 10 to the 18th. That's a pretty big number. And it's been verified up to that point. Now is this a proof? Well, the statement does restrict it to, for numbers up to this point. So for every even number below 4 times 10 to the 18th, this is a true statement. But Golbeck's conjecture says for all even integers. So after this point, after this number, right, we don't have any definite statement there yet. We do not have any guarantee for them. Even if we continue to check every even number after this number, we can only check, finally, many cases if we're trying this trial and error here. Because we only have a finite amount of time, right? It's only been a finite number of years since Golbeck's original conjecture. And so this idea of searching for constantly finding even numbers, which can be written as two primes, this will never be a valid proof unless we find one where it doesn't work. Disproving Golbeck's conjecture is arguably easier than proving it. Because to disprove it, all that we need is what we call a counter example. Because Golbeck's conjecture is about all even numbers. If I can produce for you one, just one even number for which it's not a sum of two primes, then that would give us a conjecture. And that would, sorry, that would give us a counter example to the conjecture, which would disprove it, it would not be true at that case. But this type of empirical evidence suggests that if a counter example exists, it's huge, it's really, really big. And it's reaching a point where computers are so slow at checking it. Because, I mean, after all, if you're looking for an even number bigger than this number, there's a lot of primes that you have to check, there's a lot of combinations. And so checking numbers will never, ever, ever give us a proof. But it gives us a gut feeling that the bigger these things get, that is the bigger that we've checked that we haven't found a counter example, the more evidence we have that the conjecture is probably true. But maybe, just maybe there's a point where for big numbers, because prime numbers are so sparse that it doesn't work anymore, right? Because prime numbers, you know, with these small numbers we have right here, there's a lot of small, there's a lot of small prime numbers. But when your primes get bigger and bigger and bigger, they start to get more sparse, more scarce I should say. The gaps between prime numbers can get bigger and bigger and bigger and bigger. Sort of a related problem is what's known as the twin prime conjecture. Where a twin prime are two primes that are right next to each other, like three and five, five and seven are twin primes, they're right next to each other, there's a difference of two. The twin prime conjecture, conjectures that there's an infinite amount of twin primes, that even for very large primes, you can continue to find large twin primes. And the, it's a conjecture hasn't been proven yet, but the evidence of recent years seems to suggest that the twin prime conjecture is probably true and will likely be proven within my lifetime, I would assume. There's been some major advances in that regard. Where Golbex conjecture stands, I'm not quite certain, but it could be that the scarcity of primes could happen for very large integers that reaches a point that there's not enough primes to add together and get a certain even number. It would be, that'd be very interesting if such a number exists, but it's not for certain right now. Now most mathematicians believe Golbex conjecture is in fact true, because if a counter example exists, you would think that it's been discovered by now and there's no evidence to suggest that a counter example will eventually happen. I mean it could, but we don't know. But I should mention that mathematicians with this thought process have been wrong before. It's counter examples to former conjectures that were ginormous. And so just because this number is big doesn't mean they don't happen eventually. Maybe you need numbers in the magnitude of 10 to the 30 before they start to appear. We don't know. Now the reason that the conjecture still holds is, no one's proved it yet. No one's found a counter example. No one's found a proof. But whether it's true or not actually doesn't matter too much. I mean there's going to be theoretical consequences if Golbex conjecture is true, because then we could use that to prove other things. But there also will be logical consequences if Golbex conjecture is false. And I won't dive too much into those right now as well. But what we can say is that conjectures are one of the most important parts of mathematics because they are the driving force of mathematics. In order to prove something that no one has been able to prove before, you have to do something that no one else has ever done before. New mathematics are typically built as mathematicians try to prove outstanding conjectures. There are many mathematical conjectures that exist in the present. Some of which actually have prize money. If you prove this, someone will pay you. For example, there's the famous seven millennial problems established by the Clay Institute in the year 2000, which come with a $1 million purse for anyone who can provide a valid proof, which includes counter examples if they factor false statements, to those seven problems. This video was recorded in the year 2022. And since the year 2000, I believe only one of the millennial problems has been solved, the so-called Poincaré conjecture, which is something from topology about manifolds and such. I don't think any of the other remaining six have been proven yet. If I'm correct, someone should post in the comments below. But nonetheless, these are such important problems that people are willing to pay mathematicians to solve them. And it's not necessarily the answer to the proof that matters so much, but how do people do it? Let me give you one example of a famous conjecture that has been solved, what's known as Fermat's Last Theorem. Now, this is kind of odd because it's a conjecture that actually was given the name of a theorem. And the reason it gets that name is that Pierre de Fermat in 1637 claimed that he had a proof to what's now known as Fermat's Last Theorem. And this is the famous proof that he's like, oh, the integer solutions to the equation x to the n plus y to the n equals z to the n don't exist when n is greater than 2. So they do exist like Pythagorean triples. You can find numbers like 3, 4, 5, such that 3 squared plus 4 squared equals 5 squared. But if you try to do the analogous problem for cubes or fourth powers or fifth powers, etc., you can't find integer solutions to those problems. So Fermat claimed that he had a proof to this theorem that would not fit inside the margin of the textbook he was writing at the time. Most people do not believe that Fermat had a valid proof because while this was conjectured in 1637, no one was able to provide a valid proof until like the 1990s. I don't remember the exact year. I want to say 97 or 92 or something, it was in the 1990s that Andrew Wiles finally finished the proof of Fermat's last theorem. And it wasn't that Wiles did it all by himself. This problem was worked on for like three and a half centuries where many, many mathematicians contributed to the proof of what now, what we call Fermat's last theorem right here. Wiles just happens to be the mathematician who finished it. There's no discredit to Wiles. I mean, his work was brilliant. Don't get me wrong. But there were of course dozens, maybe even hundreds of other math, brilliant mathematicians who worked on this conjecture for three and a half centuries. And the work that was done to prove Fermat's last theorem then gave it breathed life into what nowadays is called algebraic number theory and other branches of number theory. Number theory is what it is today because of Fermat's last theorem and the efforts that mathematicians took to prove it. Likewise, as mathematicians try to prove things like Golbeck's conjecture or the twin prime conjecture are the continuum hypothesis or the Riemann hypothesis. All of these statements have changed how we think about number theory or topology or set theory or so many other aspects of mathematics. The big conjectures is what drives mathematical research in the present and will do so in the future. So proper respect needs to be given to mathematical theorems. They breathe life into mathematics and math is directed by trying to prove many of these famous theorems here. And so that's going to bring us to the end of lecture five. Thanks for watching. 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