How are there unsolved math problems?
In: 9
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.
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