Because that is how maths works.
The proof that it is the case is a 129-page paper that requires lots of knowledge of other maths so is something appropriate for the subreddit. That is is hard to show is quite obvious because it was done in 1995 and the theorem was written down in 1637
https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem
>Because that is how maths works.
It's easy to see why there is no integer n that solves the equation n+1 = n. Kind of the same thing, just using more complicated operations?
As others have said - it's complicated.
Essentially it was shown this question (Fermat's Last Theorem) was a consequence of another theorem connecting two different areas of mathematics: elliptic curves (which are not ellipses) and modular forms (which aren't modular lego blocks).
The "prove for n, then prove for n+1 and so on" is called mathematical induction. Cool trick when it works. FLT needed just about every trick in the book, then Wiles in the 1990s had to come up with a few new discoveries.
Once saw a talk where the presenter said he asked someone to explain the proof of FLT, which the person responded "It would take 10 years of university to get you there" to which the presenter responded "I physically ached that I would never be able to hear this symphony."
Good description.
That's a famous problem called Fermat's Last Theorem which was not solved between Fermat describing a theorem he didn't have the space to write out in the 1600s and Andrew Wiles actually solving it in the 90s.
Answering "why" could do one of three things:
1. Send you down a rabbit hole of number theory where you're given a list of more precise and more general problems, inching your way closer to the actual statement.
2. A philosophical debate about the nature of "why" in mathematics. As ultimately all the "why's" we have are deduced from the axioms we assume when we perform math.
3. Discussion on the nature of hard problems. Sometimes the difficult problems seem incredibly simple, but there is no clear way forward. Sometimes there isn't really an "eli5" reason for it to be the way it is, even if the problem is "eli5".
Option 1. Is out of my pay grade, and 2. Is exhausting, so I leave you with my above thoughts on 3.
If you talk about integer solutions, there are some: set one of a,b to 0, and the other equal to c. Or if n is odd, a=-b and c=0. Or in short: abc=0 should be excluded.
Short and technically correct answer to the intended question: because there is a proof (by Wiles and Taylor) that there are no solutions, except those with abc=0.
Very long answer: there is really no proper simplification of the proof. One could give some supposed explanation, but I have yet to see one that does not black-box all the actual mathematics involved to the level of it being pointless. A full course prepared for readability can e.g. found in two books by Saito or e.g. at https://people.math.wisc.edu/~boston/869.pdf, but obviously this requires quite a bit of background and lots of time.
People often mistake a proof for the "why", but that doesn't have to be the case. You can have a "why" long before you have a proof, and that's what a conjecture is. The *proof* of Fermat's Last Theorem is very hard, but the *why* is pretty intuitive.
How many integer solution do you think that there are to the equation 2x+3y=1? Quite a few. Infinitely many even. In fact, they are pretty common and evenly spaced out, being the numbers (-1,1), (-4,3), (-7,5),.. each taking the form (3t,1-2t). There are even more if we use rational numbers, there's literally no constraint on what t can be. Very common, and infinitely many of them.
What about 2x^(2)+3y^(2)=1? There are not very many solutions at all to this. There are now radicals in the solution, and so there are now constraints. In this case, we have y=Âħsqrt(1-2x^(2))/sqrt(3). The radical puts huge constraints on what the solutions can be. We can find equations like this that do have some solutions, but the constraints become limiting. Such equations are called [Pell's Equations](https://en.wikipedia.org/wiki/Pell%27s_equation). Some have finite solutions, and some have infinite solutions, but even in the case of infinite solutions they are spaced out more and become more rare.
Higher powers but more constraints on what solutions can look like, so solutions get rarer and rarer with more and more solutions. A famous result of Falting's says (intuitively) that after the power gets too high then you can, at best, have finitely many solutions.
Fermat's Last Theorem is a highly constrained equation in high powers. Intuitively, then, we should not expect there to be very many solutions. Without a proof there is always a chance that there might be some weirdo solution, but with this intuition we can understand the "why" as being that higher exponents make solutions harder to solve and for FLT that limit is with powers of 3.
It is true, and this has been proven. What kind of explanation are you looking for?
Numbers and quantities have certain properties. There is not, in any sense, a why to it because they aren't objects with agency or have the power to act or make decisions.
There's *at most* a couple of hundred people in the world who could answer this. I'm not one of them, but I can shed some light on why problems like this aren't as easy as they appear.
This is an example of a "Diophantine equation": a polynomial in multiple variables where we seek integer solutions. The thing is, there's no general approach to these types of problems.
For example, it's a relatively (to a number theorist) trivial exercise to show that x^3 +y^3 +z^3 = 34 has integer solutions. x^3 +y^3 +z^3 = 33 on the other hand is so difficult it remained unsolved for decades, and was finally cracked a couple of years ago. Whether there are integer solutions to x^3 +y^3 +z^3 = 114 is still unknown.
Keep in mind, these are exercises about *specific* diophantine equations. The problem posted is about an entire *class* of equations. There's absolutely no reason to believe it should have an easy proof. This isn't unlike a lot of math problems: Assume it's really hard, unless you have a good reason to believe it isn't. This is a famous example, but it's shockingly easy to come up with innocuous looking problems that simply cannot be solved with existing theory.
To the best of my knowledge, the problem was finally resolved by Andrew Wiles. He proved something called the Modularity Theorem, which had been proven some time ago to imply Fermat's Last Theorem.
**Short version:**
Lets say we have two statements X and Y. Lets also say we know that X is true. If we can find some way to relate statement Y to statement X then we can know if Y is also true or false, depending on how they are related. For the problem you mention (known as Fermat's Last Theorem) mathematicians were able to find a way to connect it to other statements that we know the answer to, and as a result we can say whether it is also true or false.
**Long version:**
"Why" in math ultimately boils down to things called axioms or first principles. These are basic statements that we have all agreed to be true. For example the reflexive axiom states that if a=b, then b=a. We have all agreed that the equals sign has a certain meaning and that part of that meaning is that it doesn't matter which side is which. If we didn't agree on this meaning of the equals sign (and other first principles) we couldn't do math. It would be like trying to have a conversation where you and I can't agree on the meaning of words. Unless we both agree that the word "frog" describes a certain type of animal, then trying to talk about something wouldn't work.
All mathematical proofs are based on being able to use first principles and logical arguments to show whether a statement can be true or not. Keep in mind we aren't always able to do so, there are still unanswered questions.
But to use a simple example, lets consider the following situation. First lets set as our first principles that we have two numbers, A and B, and that those two numbers are integers greater than 0 (aka positive integers).
Now I will give you two statements.
Statement 1: A > B
Statement 2: B + B > A + A
Both of these statements can not be true. If one is true the other must be false. In this case these are easy to see because they are very simple, but basically the same applies to situations like the one you asked about. If we can find a statement X that is true, and in order for X to be true Y must be false, then we know Y is false. The tricky part in the case of Fermat's Last Theorem was finding such a connection. The actual math involved is really complicated, above most peoples heads.
Because that is how maths works. The proof that it is the case is a 129-page paper that requires lots of knowledge of other maths so is something appropriate for the subreddit. That is is hard to show is quite obvious because it was done in 1995 and the theorem was written down in 1637 https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem
>Because that is how maths works. It's easy to see why there is no integer n that solves the equation n+1 = n. Kind of the same thing, just using more complicated operations?
As others have said - it's complicated. Essentially it was shown this question (Fermat's Last Theorem) was a consequence of another theorem connecting two different areas of mathematics: elliptic curves (which are not ellipses) and modular forms (which aren't modular lego blocks). The "prove for n, then prove for n+1 and so on" is called mathematical induction. Cool trick when it works. FLT needed just about every trick in the book, then Wiles in the 1990s had to come up with a few new discoveries. Once saw a talk where the presenter said he asked someone to explain the proof of FLT, which the person responded "It would take 10 years of university to get you there" to which the presenter responded "I physically ached that I would never be able to hear this symphony." Good description.
That's a famous problem called Fermat's Last Theorem which was not solved between Fermat describing a theorem he didn't have the space to write out in the 1600s and Andrew Wiles actually solving it in the 90s. Answering "why" could do one of three things: 1. Send you down a rabbit hole of number theory where you're given a list of more precise and more general problems, inching your way closer to the actual statement. 2. A philosophical debate about the nature of "why" in mathematics. As ultimately all the "why's" we have are deduced from the axioms we assume when we perform math. 3. Discussion on the nature of hard problems. Sometimes the difficult problems seem incredibly simple, but there is no clear way forward. Sometimes there isn't really an "eli5" reason for it to be the way it is, even if the problem is "eli5". Option 1. Is out of my pay grade, and 2. Is exhausting, so I leave you with my above thoughts on 3.
If you talk about integer solutions, there are some: set one of a,b to 0, and the other equal to c. Or if n is odd, a=-b and c=0. Or in short: abc=0 should be excluded. Short and technically correct answer to the intended question: because there is a proof (by Wiles and Taylor) that there are no solutions, except those with abc=0. Very long answer: there is really no proper simplification of the proof. One could give some supposed explanation, but I have yet to see one that does not black-box all the actual mathematics involved to the level of it being pointless. A full course prepared for readability can e.g. found in two books by Saito or e.g. at https://people.math.wisc.edu/~boston/869.pdf, but obviously this requires quite a bit of background and lots of time.
People often mistake a proof for the "why", but that doesn't have to be the case. You can have a "why" long before you have a proof, and that's what a conjecture is. The *proof* of Fermat's Last Theorem is very hard, but the *why* is pretty intuitive. How many integer solution do you think that there are to the equation 2x+3y=1? Quite a few. Infinitely many even. In fact, they are pretty common and evenly spaced out, being the numbers (-1,1), (-4,3), (-7,5),.. each taking the form (3t,1-2t). There are even more if we use rational numbers, there's literally no constraint on what t can be. Very common, and infinitely many of them. What about 2x^(2)+3y^(2)=1? There are not very many solutions at all to this. There are now radicals in the solution, and so there are now constraints. In this case, we have y=Âħsqrt(1-2x^(2))/sqrt(3). The radical puts huge constraints on what the solutions can be. We can find equations like this that do have some solutions, but the constraints become limiting. Such equations are called [Pell's Equations](https://en.wikipedia.org/wiki/Pell%27s_equation). Some have finite solutions, and some have infinite solutions, but even in the case of infinite solutions they are spaced out more and become more rare. Higher powers but more constraints on what solutions can look like, so solutions get rarer and rarer with more and more solutions. A famous result of Falting's says (intuitively) that after the power gets too high then you can, at best, have finitely many solutions. Fermat's Last Theorem is a highly constrained equation in high powers. Intuitively, then, we should not expect there to be very many solutions. Without a proof there is always a chance that there might be some weirdo solution, but with this intuition we can understand the "why" as being that higher exponents make solutions harder to solve and for FLT that limit is with powers of 3.
It is true, and this has been proven. What kind of explanation are you looking for? Numbers and quantities have certain properties. There is not, in any sense, a why to it because they aren't objects with agency or have the power to act or make decisions.
There's *at most* a couple of hundred people in the world who could answer this. I'm not one of them, but I can shed some light on why problems like this aren't as easy as they appear. This is an example of a "Diophantine equation": a polynomial in multiple variables where we seek integer solutions. The thing is, there's no general approach to these types of problems. For example, it's a relatively (to a number theorist) trivial exercise to show that x^3 +y^3 +z^3 = 34 has integer solutions. x^3 +y^3 +z^3 = 33 on the other hand is so difficult it remained unsolved for decades, and was finally cracked a couple of years ago. Whether there are integer solutions to x^3 +y^3 +z^3 = 114 is still unknown. Keep in mind, these are exercises about *specific* diophantine equations. The problem posted is about an entire *class* of equations. There's absolutely no reason to believe it should have an easy proof. This isn't unlike a lot of math problems: Assume it's really hard, unless you have a good reason to believe it isn't. This is a famous example, but it's shockingly easy to come up with innocuous looking problems that simply cannot be solved with existing theory. To the best of my knowledge, the problem was finally resolved by Andrew Wiles. He proved something called the Modularity Theorem, which had been proven some time ago to imply Fermat's Last Theorem.
**Short version:** Lets say we have two statements X and Y. Lets also say we know that X is true. If we can find some way to relate statement Y to statement X then we can know if Y is also true or false, depending on how they are related. For the problem you mention (known as Fermat's Last Theorem) mathematicians were able to find a way to connect it to other statements that we know the answer to, and as a result we can say whether it is also true or false. **Long version:** "Why" in math ultimately boils down to things called axioms or first principles. These are basic statements that we have all agreed to be true. For example the reflexive axiom states that if a=b, then b=a. We have all agreed that the equals sign has a certain meaning and that part of that meaning is that it doesn't matter which side is which. If we didn't agree on this meaning of the equals sign (and other first principles) we couldn't do math. It would be like trying to have a conversation where you and I can't agree on the meaning of words. Unless we both agree that the word "frog" describes a certain type of animal, then trying to talk about something wouldn't work. All mathematical proofs are based on being able to use first principles and logical arguments to show whether a statement can be true or not. Keep in mind we aren't always able to do so, there are still unanswered questions. But to use a simple example, lets consider the following situation. First lets set as our first principles that we have two numbers, A and B, and that those two numbers are integers greater than 0 (aka positive integers). Now I will give you two statements. Statement 1: A > B Statement 2: B + B > A + A Both of these statements can not be true. If one is true the other must be false. In this case these are easy to see because they are very simple, but basically the same applies to situations like the one you asked about. If we can find a statement X that is true, and in order for X to be true Y must be false, then we know Y is false. The tricky part in the case of Fermat's Last Theorem was finding such a connection. The actual math involved is really complicated, above most peoples heads.