How are there unsolved math problems?
In: 9
Imagine this
You know there’s a certain number of countable things in your room between socks tshirts and pillows and everything else ,but if i asked you to count everything in your room you’d have to figure out what to count as a thing ,cause not everything is a thing ,and you’d face a case of an unsolvable problem ,we know there’s a solution that exists ,but it’s so mind boggling hard to calculate that we ,lazy humans just leave it be for now
They’re not “math problems” in the school sense…it’s not like there’s an equation that we just can’t solve, although there are some equations so complicated that we can only solve them numerically on a computer, not with regular algebra.
“Unsolved math problems” are statements about mathematics that we (so far) can’t prove are true or false. For example, until pretty recently, we didn’t know if there were any integer solutions to the equation “x^n + y^n = z^n” for any n bigger than 2. It sure looked like there weren’t, we couldn’t find any, but just because we can’t find it doesn’t mean it doesn’t exist. Infinitiy is annoying that way. And mathmaticians *hate* ambiguity…they want to prove that something either is definitely true or definitely isn’t. This particular problem was called “Fermat’s Last Theorem”. Fermat was a mathmatician who wrote it down (the problem) in 1637 but we didn’t figure out how to prove it until 1993/1994. It was unsolved for over 300 years.
Other unsolved math problems are like that…things we’re not sure are true or not…they look true but we can’t prove it and, until we prove it (or not) it will remain “unsolved”.
Most of the things that we consider unsolved math problems are _conjectures_. That basically means they’re educated guesses about patterns or relationships (between numbers) that may or may not exist. However, just because someone guesses that this relationship may exist doesn’t mean it’s proven to exist. And the guess itself may not provide strong guidance on how to prove it.
For example, I may say to you that I conjecture that for _every_ prime number you find, I can find a bigger one. This would be considered unsolved until someone either proved me right or wrong.
There is actually a list of math problems that you get money (1million dollars) for if you can solve them. https://en.m.wikipedia.org/wiki/Millennium_Prize_Problems
Explaining this to you as though you were five requires having known you when you were five.
We csn only approximate your mentality at that age to give tou an explanation that we guess you would accept at that age.
So knowing exactly how to explain this to you as though you were five is impossible because there are too many variables of unknown quantities describing approximations of appropriage qualities.
The only way to explain it is through approximations because the real results are unknowable through math.
Because a simple problem can be very hard to solve. The type of problem are in general not finding a single solution to a problem like in school, It is more to show of something that is general for all number or something impossible for any number.
An example that is simple to show is that the sum of two odd integers is always a even integer
To do that you can just test all possibilities because there is an infinite number of them. You need to use a general expression that is valid for all odd and even numbers to solve the problem.
the solution is an even integer n can be written as n=2k where k is an integer. An odd integer is n=2k+1 where k is an integer
So let 2A+1 and 2B+1 be our odd intetegers. The sum is 2A+1 +2B+1 =2A+2B+2
If we factor out the 2 we get 2(A+B+1). A+B+1 is an integer that we call D. Now we got the sum as 2D where D is an integer. That how to see describe an even integer above. So the sum of any two odd integer is a even integer
This is e example of the type of maths that can have unsolvable problems.
An example of a solved problem that took a long time is [Fermat’s Last Theorem](https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem) written down in 1637
The theorem is: Theorem three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.
It was written in the margin of a book and Fermat also added that he had proof that was too large to fit in the margin. We are quite sure that he did not have a solution to the problem. He likely taught he did but was likely mistaken
It is easy to show that if n is 1 or 2 there is a solution.
For n=1 a=1, b=1 and c=2 is a solution. because 1^1 + 1^1 =2^1 is equal to 1+1=2
For n=2 a=3, b=4 and c=5 is a solution 3^2 + 4^2 = 5^2 is equal to 9 + 16 =25
Now the problem is how do you show that there is no solution for n=3 or larger? You can test it with n=3 and a, b, and c smaller than 1 trillion but there might be a solution where a is a number with 1 million digits.
Even if you find a way to show it’s impossible for any and if n=3 you have to do it for any n. There might be a solution where n has 1 trillion digits. Testing all possibilities is impossible.
It was solved in two papers with a total of 129 pages in 1995. It used lots of maths developed in the day of Fermat. What this show is that a problem that can be stated in a single line can be extremely hard to sovle.
Here’s a simple one to understand:
Pick a number (non-negative, no decimals).
1. If it’s even, divide it by half.
2. If it’s odd, multiply it by 3, then add 1.
Keep repeating these steps with the new numbers you get.
You’ll notice that once you reach 4, the next numbers in the sequence are 2, then 1, then 4, then 2, then 1, and so on. From there, it loops forever.
**The question is:** will you always reach this loop with any starting number you pick?
This is called the [Collatz conjecture](https://en.wikipedia.org/wiki/Collatz_conjecture) and has remained unsolved for almost 100 years.
These are what unsolved math problems look like. They’re not equations that need solving, they ask a question which needs an answer and a proof of the answer.
Many unsolved maths problems are general-type statements of the form:
“For all known integers, this rule/scenario applies….”
“For all known examples of [scenario] there is/is not a solution that fits this pattern….”
“There is no integer solution to [extremely complex and detailed equation]..”
etc
These are things that can be very difficult to prove, because the set of numbers is infinite. If we haven’t found an solution to a scenario with all the numbers we’ve tried, that doesn’t necessarily mean there’s no solution, it’s just that it hasn’t been found yet. It’s entirely possible some enormous or specific number will indeed fit the solution. Similarly, just because every integer we’ve tested satisfies a pattern/rule/scenario, doesn’t necessarily mean that every single possible integer in existence will as well.
So it’s not like people can just brute force or trial-and-error a solution (x + 1 = 3….does it work if x=1? No. Does it work if x=2? Aha!) To solve these problems, they would somehow need to construct a general rule, and prove that it can work for all numbers or all scenarios outlined in the problem.
Most proofs in math start with a conjecture. Some mathematician notices some pattern with a small set of numbers, then tests that pattern over more and more numbers. At this point they are just looking for a number or numbers that violate the pattern to prove the pattern doesn’t exist.
However, there are an infinite number of numbers so they can’t prove the pattern applies to every number.
Instead, proofs must cleverly combine what is known or proven already to prove the pattern.
For example, if you can prove that all bears are brown and prove that all grizzlies are bears, you can now make a proof saying that all grizzlies are brown.
But the proofs are much more complicated than this and there are many, many proofs you can use as steps in proving your conjecture. In our example, we could have a proof that proves that all fish swim. However, that proof doesn’t help prove a grizzly is brown… Or could it?
The sheer massive amount of proofs that can be combined and found to be applicable to each other in clever ways is nearly infinite too, so it’s not always easy to prove new things using old proofs.
Interestingly, people have started using computers to combine random proofs together to try and prove new things but this is obviously quite tricky to encode.
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