This theorem is so famous that even people who don’t know what math is have heard of it. But let’s not get ahead of ourselves, alright?
To begin with: What the ***** is FLT? Well, imagine you’re playing a game with your friends where you pick two numbers and multiply them together. Then, you add those two numbers to each other and raise that sum to some power (let’s say 3 for now). If the result of this calculation is equal to one of the original numbers, then congratulations! You have just discovered a counterexample to FLT.
Now, let me explain what I mean by “counterexample.” In math, we love proving things and finding patterns that work all the time. But sometimes, there are exceptions situations where our pattern doesn’t hold true. These exceptions are called counterexamples. And in this case, if you can find a pair of numbers that satisfy FLT’s conditions (i.e., they multiply to give another number and then adding them together raised to the power 3 equals one of those original numbers), then we know that FLT is wrong.
But here’s the kicker: nobody has ever found such a counterexample! In fact, mathematicians have been trying for over 400 years to prove or disprove FLT, and as far as we know today, it remains an unsolved problem in math.
So why is this theorem so important? Well, first of all, it’s a beautiful piece of mathematics that has fascinated people for centuries. But more importantly, it’s a testament to the power of human curiosity and determination. For over 400 years, mathematicians have been trying to solve FLT, and while they haven’t succeeded yet, their efforts have led to some incredible discoveries in other areas of math.
In fact, one of the most famous proofs in all of mathematics Andrew Wiles’ proof of Fermat’s Last Theorem for n = 3 (which we’ll talk about more later) was a direct result of trying to solve FLT. And while that proof didn’t actually prove FLT itself, it did open up new avenues of research and led to some amazing breakthroughs in other areas of math.
So what are the conditions for FLT? Well, let me explain them using an example:
Suppose we choose two numbers say 5 and 7. We multiply those together to get 35. Then, we add those two numbers (5 + 7) to get 12. Finally, we raise that sum to the power of 3 (i.e., 12^3 = 1728). If any one of these steps results in a number other than one of our original numbers (in this case, neither 5 nor 7), then FLT doesn’t apply and we can move on to another pair of numbers.
But if all three steps result in the same number as one of our original numbers (i.e., either 35 or 12^3 is equal to 5 or 7), then we have a potential counterexample to FLT! And that’s what mathematicians have been trying to find for over 400 years a pair of numbers that satisfy these conditions and prove that FLT is wrong.
So why hasn’t anyone found such a counterexample yet? Well, there are a few reasons: firstly, the number of possible pairs of numbers is infinite (i.e., we can choose any two integers greater than 1), so it would take an incredibly long time to check them all. Secondly, even if we could somehow magically find every pair of numbers and calculate whether they satisfy FLT’s conditions or not, the calculations involved are extremely complex and require some serious math skills (which is why mathematicians have been working on this problem for so long).
But despite these challenges, there has been some progress in recent years. In 1995, Andrew Wiles finally proved FLT for n = 3 that is, he showed that if we choose any three numbers (not necessarily integers) and multiply them together to get a fourth number, then raising the sum of those first three numbers to the power of 3 will never result in one of our original numbers.
This proof was a major breakthrough in math not just because it solved FLT for n = 3 (which is still an incredible achievement), but also because it opened up new avenues of research and led to some amazing discoveries in other areas of math. In fact, Wiles’ proof has been called one of the most important mathematical achievements of all time!
So what does this mean for us? Well, firstly, it means that we can finally put FLT to rest (at least for n = 3) and move on to other problems in math. But more importantly, it shows us just how powerful human curiosity and determination can be even when faced with seemingly insurmountable challenges, we can still push forward and make incredible breakthroughs that change the course of history.