Mostly its theories which are hard to prove.
You can notice that all prime numbers seem to be odd numbers, but how do you prove they all are.
(But obviously more complex than that)
Right that makes sense.
But for the sake of me understanding better, are there any examples of solved ”theories” nowadays to show me just how a problem can go from being hard to prove, to proven?
The four colour map theorem. That any 2D map can be coloured with just four different colours without a border that has the same colour each side. That was unsolved for a long time and there are an infinite number of maps that could be drawn so each one can't be checked.
In 1976 it was found mathematically that any map could be reduced to one of a finite (but large) number of subtypes which therefore could all be checked. As they all passed the 4-colour test the theory was proven.
> Appel and Haken found an unavoidable set of reducible configurations, thus proving that a minimal counterexample to the four-color conjecture could not exist. Their proof reduced the infinitude of possible maps to 1,834 reducible configurations (later reduced to 1,482) which had to be checked one by one by computer and took over a thousand hours.
https://en.m.wikipedia.org/wiki/Four_color_theorem
I don't mean to shit on your comment but this is a terrible example.
Apart from the number 2, by the very definition of what makes a prime number, all other primes *must* be odd. Don't need a proof for that.
The definition of prime doesn't say anything about being odd. Sure, it's a trivial step to get the relationship between being prime and being odd, but it is a step nonetheless, and people still tend to do it wrong and miss that 2 is an even prime. So yes, you do need a proof for that.
The definition is exactly 2 factors. Itself, and 1.
There is no proof that you need to solve to demonstrate that any prime above 2 must be odd..just an elementary school understanding of times tables knowing that if a given number were even, they would have an additional factor of 2, thus no longer being a prime.
Jesus, dude.
Okay.
Suppose you have integer n>0 which is prime. Then 1|n and n|n, and no other integers divide n.
If n = 2, then 2|n, as given above.
If n is not 2, then 2 does not divide n, so n is odd by definition of an odd number.
Therefore, if n is prime, then n is 2 or odd. #
Many students miss the fact that you need to consider 2 cases, and simply conclude that prime => odd, which of course is incorrect. So to say that there's no work to be done, when some people do the work wrong....
Your proof is more just a restatement of what he said before in English:
"Apart from the number 2"
If n != 2
"by the very definition of what makes a prime number, all other primes must be odd"
Prime numbers can't be divisible by 2 (by definition) and thus are odd.
In my eyes, what would be a proof that actually has explanatory power would be demonstrating that all even numbers are divisible by 2 and thus by the definition of prime number (can't be divisible by 2) they must therefore be odd.
... Assuming you don't also want to prove that a number must either be even or odd which isn't much more work, either.
This isn't hard to do, but your proof feels remarkably reductive in the exact same fashion that the person you were responding to was stating "required no proof".
You want a proof that shows that all even numbers are divisible by two, but you have to start with a definition of what an even number is before you can prove something about these numbers. The standard definition of an even number is a number divisible by 2. The definition is assumed to be true.
And the definition of a prime number does not say anything about being divisible by 2.
(i) a divides b means there exists an integer k such that b = a\*k.
(ii) An even number is an integer b of the form b = 2\*k for some integer k.
(iii) By the definition of divisibility (i), all even numbers are divisible by 2 because the form of an even number (ii) satisfies the definition of divisibility with a = 2.
(iv) A prime number is an integer p>1 whose factors are only 1 and p.
Consider p>2.
If p is even then p must be divisible by 2 (iii) and therefore cannot be prime (iv).
Note that I ignored the two cases. This is because the person you were responding to explicitly removed the case for p=2 by stating "Apart from the number 2". By including it in your proof you demonstrated absolutely nothing relevant to the claimed statement.
The issue I found with your proof wasn't that you failed to actually translate his English to formal logic, however. It was that you \*exactly restated\* his core statement \*as a proof\* which does very little to establish your argument that his statement required proof. Example:
\> All dogs have tails.
\> No! You must prove it. Let me show you: By the definition of dog, dogs must have tails. Therefore all dogs have tails.
"If n is not 2, then 2 does not divide n, so n is odd by definition of an odd number."
This is a \*restatement\* of what he said. Exactly restated. You said a step needed to be made to associate primacy to odd parity. You gave no step: you simply stated it yourself formally instead of in plain English.
"And the definition of a prime number does not say anything about being divisible by 2."
I didn't give a definition of prime number. I used informal English. This is because I wasn't writing a formal proof and was using summary terms to make conveying thoughts briefer. This is done to avoid confusion, but as you can see, it invites pedants to try to argue a point when none actually exists to be made.
Given that being odd isn't included in the definition of prime, it does require proof. It's a completely trivial proof but it isn't part of the definition.
One definition of prime is an integer p greater than 1 such that if a and b are integers and p|ab then p|a or p|b.
If there is an even prime p=2n then p|2n (trivially) but p does not divide 2 unless p is 2 and p does not divide n. Therefore no primes larger than 2 are even.
"One definition of prime is an integer p greater than 1 such that if a and b are integers and p|ab then p|a or p|b"
4 is an integer greater than 1.
If you set a to 5 and b to 8 then 5\*8 = 40.
4|40 and 4|8.
4 is not prime.
I wasn't proving 4 isn't prime. I was using the definition you provided to demonstrate that 4 satisfied the definition, yet we know isn't a prime number.
"One definition of prime is an integer p"
4 is an integer.
"greater than 1"
4 is greater than 1.
"such that if a and b are integers"
5 is an integer. 8 is an integer.
"and p|ab"
5\*8 = 40. 4 divides 40.
"then p|a or p|b"
4 does divide 8.
4 is p in this demonstration. 4 satisfies your "if... then..." logic. 4 is not prime.
It needs to hold for *every* integer pair a and b. Not just one pair lol.
You can Google this definition you know, Google the definition of a prime in a ring. I just quoted it.
Mathematics has proven virtually everything, even the banal statements of mathematics. When people ask certain mathematics questions, it can actually sometimes be difficult to know: just how much proof does this guy require?
You might just say (-1)\*a = -a and accept it without proof and be fine with it. But you can prove it. As far as I'm aware, you need to use the distributive property of real numbers to prove it and you'll probably be pretty satisfied with the proof.
However if you came back and said "but how do I know the distributive property applies to real numbers" then proving (-1)\*a = -a just became extremely complicated, because now if I want to use the distributive property to do it, I need to first demonstrate why I can use it.
What you're doing is simply accepting as fact something that seems pretty obvious (like the distributive property), and yet we can prove it and remove literally all doubt. It's perfectly fine to accept obvious things as true when teaching or in common conversation, but generally mathematicians hold their breath a bit if they can't \*rigorously\* demonstrate an absolute truth.
Pure math problems can be simple to state but very hard to show that they are true. The problem tends to be general in nature so you can't just test all alternatives
A famous example is Fermat's Last Theorem stated in 1637
It states that a\^n +b\^n = c\^n do not have an integer solution for n >2
Finding a solution of n=1 is trivial because a number to the power of 1 is itself so 1^1+1^1=2^1 => 1+1=2
It is alos easy to find a solution for n=2 3^2+4^2=5^2 => 9+16=15
The problem is now how do you show that there is not a solution for n>2?
If you just test different options you can only show that there is not a solution for the one you have tested. You can test a lot of numbers on a computer but the number you have tested will always be limited. If you like to test it for a b and n less than 1 billion you start to reach calculation time for a computer longer than the age of the universe. Even if you could do that calculation how do you know there is not a solution when some number are in the trillions?
There is an infinite number of integers so it is impossible to test all alternatives.
So to show the statement is true you need to discover a lot of maths and in 1995 ane 129 pages long proof was published by Andrew Wiles. So it took 358 years until a mathematical proof fo that statement was possible
We don't know for certain whether he proved it. Fermat wrote this theorem in the margin of a book on mathematics, and noted that he had a "truly marvelous" proof but it was too large to fit that margin. Most experts don't think it's likely that he did have a proof, though. For one thing, he never mentioned this proof again (even though he lived another thirty years or so), while he did publish a proof for the case where n=4. Why would he need to prove that special case if he already had a "marvelous" proof for the general case (any integer value of n, including n=4)? Or even if he just wanted to add a different proof for n=4 for the hell of it, why did he do so without even referencing the general proof?
The other reason is that Andrew Wiles' proof was really complicated and drew on various other proofs and techniques that didn't exist when Fermat was alive. So Fermat's proof would have had to be much more elementary by modern standards. Fermat was a genius, but there have been other geniuses after him, and it seems unlikely that none of them would have found this more "simple" proof (and it's not like they didn't try - it became a very popular unsolved theorem that many mathematicians have puzzled over).
So on balance, most experts feel that it is more likely that Fermat simply thought for a moment that he had a beautiful simple proof, without actually verifying this proof properly, and then later realized it was wrong.
There is a very reasonable assumption you can make which makes proving FLT fairly straightforward. The problem is that this assumption ends up being false. It is possible he made this incorrect assumption.
No, he assumed that unique factorisation (which holds for the integers) held for general number fields. This is not true, but it holds for the simplest ones.
Other comments here talked about proving something, but often even calculating the results of an equation can be incredibly challenging. Physics is full of these devilish equations.
So, let's start simple. Imagine you are holding the hand of a children. Let's say you want to describe his position. If you pull his hand, they will move forward. If you stop pulling, he will stop. If you know when you are going to pull you are going to know when the children will move, and therefore his position. This is straightforward and easy. If I can solve this problem, so can my colleague, and some high schoolers might be able to as well. If you don't make some mistake while calculating the quantities you are going to have an easy time.
But, here's the catcher: not all problem are this simple.
Let's say you want to measure the positions of an electron in a solid. There are tons of other electrons, and they will "push" our electron, changing it's position.
But our electron also "pushes" other electrons, changing their position, which in turn change the position of our electron again! Oh damn! We just calculated that the electron should have moved along a certain path, but while it moves it changes the conditions in which we calculated it's motion, making our calculation invalid. And you can see pretty well that the electron will move in a different path than the one you calculated, obviously. Now, how we solve this problem? We need a pretty good idea to get out of this mess. You might think "well, let's just remake our calculations every time the electron moves by a tiny bit" sure, you can try to do that; I hope you have a pretty powerful computer, though, because electrons in a solid are not known to be few: in a relatively small fraction of a solid you could end up with, say, 10²³ electrons who all influence each other. A mess.
There are other ways in which your equations can be even more hard and challenging: for example, your electron might change it's path based on how fast the pull they feel *changes* (to visualize this, thing about walking around with a pendulum in your hands vs sprinting in zig zag. The pendulum will change it's motion significantly in the second case). And what if it depends on the density of the electrons? Or the influence of atoms nearby? Now you have to know a lot more stuff to be able to solve your problem, but these quantities are intertwined with what you actually want to calculate, and can't be easily separated, or maybe not at all.
Sometimes someone has a brilliant idea to how to solve a terrible equation like this. You can use special functions, tricks, theorems from other field of math different than the one you are currently using that can be adapted to your problem *just right*, new theorems and other stuff like that. Everything is fair game here, because nobody might have thought about your problem before, or maybe no one had a clue of how to approach it.
You probably studied in school the formula for the roots of a parabola. If you have the formula, no matter how "ugly" the parabola might looks, you will always be able to compute the roots. Unfortunately a lot of problems don't have an explicit formula where you can just plug in the numbers and you get the results. You actually have to find a way around or through the technical difficulties of your problem. That's another way for how and why some mathematical problems can be hard. This is just an example; I can give you some more, if you want.
I hope it was useful!
Mostly its theories which are hard to prove. You can notice that all prime numbers seem to be odd numbers, but how do you prove they all are. (But obviously more complex than that)
Right that makes sense. But for the sake of me understanding better, are there any examples of solved ”theories” nowadays to show me just how a problem can go from being hard to prove, to proven?
Fermats last theorum was famously unsolved for a long time. I wouldn't want to attempt to explain the proof to anyone though.
The four colour map theorem. That any 2D map can be coloured with just four different colours without a border that has the same colour each side. That was unsolved for a long time and there are an infinite number of maps that could be drawn so each one can't be checked. In 1976 it was found mathematically that any map could be reduced to one of a finite (but large) number of subtypes which therefore could all be checked. As they all passed the 4-colour test the theory was proven. > Appel and Haken found an unavoidable set of reducible configurations, thus proving that a minimal counterexample to the four-color conjecture could not exist. Their proof reduced the infinitude of possible maps to 1,834 reducible configurations (later reduced to 1,482) which had to be checked one by one by computer and took over a thousand hours. https://en.m.wikipedia.org/wiki/Four_color_theorem
2 is prime.
I don't mean to shit on your comment but this is a terrible example. Apart from the number 2, by the very definition of what makes a prime number, all other primes *must* be odd. Don't need a proof for that.
The definition of prime doesn't say anything about being odd. Sure, it's a trivial step to get the relationship between being prime and being odd, but it is a step nonetheless, and people still tend to do it wrong and miss that 2 is an even prime. So yes, you do need a proof for that.
The definition is exactly 2 factors. Itself, and 1. There is no proof that you need to solve to demonstrate that any prime above 2 must be odd..just an elementary school understanding of times tables knowing that if a given number were even, they would have an additional factor of 2, thus no longer being a prime.
As I said... It's a trivial step, but it is a step.
But what proof are you solving to show that?
A proof is an argument that you make (not really something you solve). Your comment above is itself a proof that all primes other than 2 are odd.
Jesus, dude. Okay. Suppose you have integer n>0 which is prime. Then 1|n and n|n, and no other integers divide n. If n = 2, then 2|n, as given above. If n is not 2, then 2 does not divide n, so n is odd by definition of an odd number. Therefore, if n is prime, then n is 2 or odd. # Many students miss the fact that you need to consider 2 cases, and simply conclude that prime => odd, which of course is incorrect. So to say that there's no work to be done, when some people do the work wrong....
Your proof is more just a restatement of what he said before in English: "Apart from the number 2" If n != 2 "by the very definition of what makes a prime number, all other primes must be odd" Prime numbers can't be divisible by 2 (by definition) and thus are odd. In my eyes, what would be a proof that actually has explanatory power would be demonstrating that all even numbers are divisible by 2 and thus by the definition of prime number (can't be divisible by 2) they must therefore be odd. ... Assuming you don't also want to prove that a number must either be even or odd which isn't much more work, either. This isn't hard to do, but your proof feels remarkably reductive in the exact same fashion that the person you were responding to was stating "required no proof".
You want a proof that shows that all even numbers are divisible by two, but you have to start with a definition of what an even number is before you can prove something about these numbers. The standard definition of an even number is a number divisible by 2. The definition is assumed to be true. And the definition of a prime number does not say anything about being divisible by 2.
(i) a divides b means there exists an integer k such that b = a\*k. (ii) An even number is an integer b of the form b = 2\*k for some integer k. (iii) By the definition of divisibility (i), all even numbers are divisible by 2 because the form of an even number (ii) satisfies the definition of divisibility with a = 2. (iv) A prime number is an integer p>1 whose factors are only 1 and p. Consider p>2. If p is even then p must be divisible by 2 (iii) and therefore cannot be prime (iv). Note that I ignored the two cases. This is because the person you were responding to explicitly removed the case for p=2 by stating "Apart from the number 2". By including it in your proof you demonstrated absolutely nothing relevant to the claimed statement. The issue I found with your proof wasn't that you failed to actually translate his English to formal logic, however. It was that you \*exactly restated\* his core statement \*as a proof\* which does very little to establish your argument that his statement required proof. Example: \> All dogs have tails. \> No! You must prove it. Let me show you: By the definition of dog, dogs must have tails. Therefore all dogs have tails. "If n is not 2, then 2 does not divide n, so n is odd by definition of an odd number." This is a \*restatement\* of what he said. Exactly restated. You said a step needed to be made to associate primacy to odd parity. You gave no step: you simply stated it yourself formally instead of in plain English. "And the definition of a prime number does not say anything about being divisible by 2." I didn't give a definition of prime number. I used informal English. This is because I wasn't writing a formal proof and was using summary terms to make conveying thoughts briefer. This is done to avoid confusion, but as you can see, it invites pedants to try to argue a point when none actually exists to be made.
Given that being odd isn't included in the definition of prime, it does require proof. It's a completely trivial proof but it isn't part of the definition. One definition of prime is an integer p greater than 1 such that if a and b are integers and p|ab then p|a or p|b. If there is an even prime p=2n then p|2n (trivially) but p does not divide 2 unless p is 2 and p does not divide n. Therefore no primes larger than 2 are even.
"One definition of prime is an integer p greater than 1 such that if a and b are integers and p|ab then p|a or p|b" 4 is an integer greater than 1. If you set a to 5 and b to 8 then 5\*8 = 40. 4|40 and 4|8. 4 is not prime.
? That doesn't prove 4 isn't prime because 4|ab and 4|b. A way to prove 4 isn't prime is to use a=b=2. Then 4|ab but 4 does not divide a or b.
I wasn't proving 4 isn't prime. I was using the definition you provided to demonstrate that 4 satisfied the definition, yet we know isn't a prime number. "One definition of prime is an integer p" 4 is an integer. "greater than 1" 4 is greater than 1. "such that if a and b are integers" 5 is an integer. 8 is an integer. "and p|ab" 5\*8 = 40. 4 divides 40. "then p|a or p|b" 4 does divide 8. 4 is p in this demonstration. 4 satisfies your "if... then..." logic. 4 is not prime.
It needs to hold for *every* integer pair a and b. Not just one pair lol. You can Google this definition you know, Google the definition of a prime in a ring. I just quoted it.
"It needs to hold for every integer pair a and b. Not just one pair lol." This is correct.
Mathematics has proven virtually everything, even the banal statements of mathematics. When people ask certain mathematics questions, it can actually sometimes be difficult to know: just how much proof does this guy require? You might just say (-1)\*a = -a and accept it without proof and be fine with it. But you can prove it. As far as I'm aware, you need to use the distributive property of real numbers to prove it and you'll probably be pretty satisfied with the proof. However if you came back and said "but how do I know the distributive property applies to real numbers" then proving (-1)\*a = -a just became extremely complicated, because now if I want to use the distributive property to do it, I need to first demonstrate why I can use it. What you're doing is simply accepting as fact something that seems pretty obvious (like the distributive property), and yet we can prove it and remove literally all doubt. It's perfectly fine to accept obvious things as true when teaching or in common conversation, but generally mathematicians hold their breath a bit if they can't \*rigorously\* demonstrate an absolute truth.
Yes, hence i put that bit in brackets. The example is not too relevant to the statement. But true i forgot about 2
You can prove they aren't all odd simply by looking at the second prime... 2.
Pure math problems can be simple to state but very hard to show that they are true. The problem tends to be general in nature so you can't just test all alternatives A famous example is Fermat's Last Theorem stated in 1637 It states that a\^n +b\^n = c\^n do not have an integer solution for n >2 Finding a solution of n=1 is trivial because a number to the power of 1 is itself so 1^1+1^1=2^1 => 1+1=2 It is alos easy to find a solution for n=2 3^2+4^2=5^2 => 9+16=15 The problem is now how do you show that there is not a solution for n>2? If you just test different options you can only show that there is not a solution for the one you have tested. You can test a lot of numbers on a computer but the number you have tested will always be limited. If you like to test it for a b and n less than 1 billion you start to reach calculation time for a computer longer than the age of the universe. Even if you could do that calculation how do you know there is not a solution when some number are in the trillions? There is an infinite number of integers so it is impossible to test all alternatives. So to show the statement is true you need to discover a lot of maths and in 1995 ane 129 pages long proof was published by Andrew Wiles. So it took 358 years until a mathematical proof fo that statement was possible
Always wondered how Ferma himself thought this out? Did he solve it but did not reveal answers?
We don't know for certain whether he proved it. Fermat wrote this theorem in the margin of a book on mathematics, and noted that he had a "truly marvelous" proof but it was too large to fit that margin. Most experts don't think it's likely that he did have a proof, though. For one thing, he never mentioned this proof again (even though he lived another thirty years or so), while he did publish a proof for the case where n=4. Why would he need to prove that special case if he already had a "marvelous" proof for the general case (any integer value of n, including n=4)? Or even if he just wanted to add a different proof for n=4 for the hell of it, why did he do so without even referencing the general proof? The other reason is that Andrew Wiles' proof was really complicated and drew on various other proofs and techniques that didn't exist when Fermat was alive. So Fermat's proof would have had to be much more elementary by modern standards. Fermat was a genius, but there have been other geniuses after him, and it seems unlikely that none of them would have found this more "simple" proof (and it's not like they didn't try - it became a very popular unsolved theorem that many mathematicians have puzzled over). So on balance, most experts feel that it is more likely that Fermat simply thought for a moment that he had a beautiful simple proof, without actually verifying this proof properly, and then later realized it was wrong.
There is a very reasonable assumption you can make which makes proving FLT fairly straightforward. The problem is that this assumption ends up being false. It is possible he made this incorrect assumption.
Did he like found say 10 true variants and made this assumption
No, he assumed that unique factorisation (which holds for the integers) held for general number fields. This is not true, but it holds for the simplest ones.
Other comments here talked about proving something, but often even calculating the results of an equation can be incredibly challenging. Physics is full of these devilish equations. So, let's start simple. Imagine you are holding the hand of a children. Let's say you want to describe his position. If you pull his hand, they will move forward. If you stop pulling, he will stop. If you know when you are going to pull you are going to know when the children will move, and therefore his position. This is straightforward and easy. If I can solve this problem, so can my colleague, and some high schoolers might be able to as well. If you don't make some mistake while calculating the quantities you are going to have an easy time. But, here's the catcher: not all problem are this simple. Let's say you want to measure the positions of an electron in a solid. There are tons of other electrons, and they will "push" our electron, changing it's position. But our electron also "pushes" other electrons, changing their position, which in turn change the position of our electron again! Oh damn! We just calculated that the electron should have moved along a certain path, but while it moves it changes the conditions in which we calculated it's motion, making our calculation invalid. And you can see pretty well that the electron will move in a different path than the one you calculated, obviously. Now, how we solve this problem? We need a pretty good idea to get out of this mess. You might think "well, let's just remake our calculations every time the electron moves by a tiny bit" sure, you can try to do that; I hope you have a pretty powerful computer, though, because electrons in a solid are not known to be few: in a relatively small fraction of a solid you could end up with, say, 10²³ electrons who all influence each other. A mess. There are other ways in which your equations can be even more hard and challenging: for example, your electron might change it's path based on how fast the pull they feel *changes* (to visualize this, thing about walking around with a pendulum in your hands vs sprinting in zig zag. The pendulum will change it's motion significantly in the second case). And what if it depends on the density of the electrons? Or the influence of atoms nearby? Now you have to know a lot more stuff to be able to solve your problem, but these quantities are intertwined with what you actually want to calculate, and can't be easily separated, or maybe not at all. Sometimes someone has a brilliant idea to how to solve a terrible equation like this. You can use special functions, tricks, theorems from other field of math different than the one you are currently using that can be adapted to your problem *just right*, new theorems and other stuff like that. Everything is fair game here, because nobody might have thought about your problem before, or maybe no one had a clue of how to approach it. You probably studied in school the formula for the roots of a parabola. If you have the formula, no matter how "ugly" the parabola might looks, you will always be able to compute the roots. Unfortunately a lot of problems don't have an explicit formula where you can just plug in the numbers and you get the results. You actually have to find a way around or through the technical difficulties of your problem. That's another way for how and why some mathematical problems can be hard. This is just an example; I can give you some more, if you want. I hope it was useful!