"Well, some mathematics problems look simple, and you try them for a year or so, and then you try them for a hundred years, and it turns out that they're extremely hard to solve. There's no reason why these problems shouldn't be easy, and yet they turn out to be extremely intricate. The Last Theorem is the most beautiful example of this." - Andrew Wiles
One of the greatest mathematical minds in history left us on this day in 1665. But Pierre de Fermat wasn't quite done with us yet. You see, an innocent little inscription he left behind in one of his notebooks proceeded to baffle and challenge the best minds in mathematics for more than 300 years. Here it is:
"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain."
The ancient Pythagorean Theorem says that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. In easier to understand algebra, that looks like this:
x^2 + y^2 = z^2
And example would be: 5^2 + 12^2 = 13^2, but there are many more.
Know it or not, we use this every day. It's one of the most fundamental and important relationships in all of geometry. We couldn't design a thing without it. Even grade school kids learn this.
But what Fermat was saying is that for any power of this equation higher than 2, there is no whole number solution. So don't look for it. An example would be:
x^5 + y^5 = z^5
According to Fermat, this has no whole number solution, and it never will. And he can prove it. In fact he DID prove it. He just ran out of room in the margins of his notebook and so he left it to us to re-discover.
Now, I don't know about you, but if someone told me they had solved a very difficult problem but couldn't show you because they ran out of paper, I might be skeptical. And many people were. They said he was either messing with us, or he didn't know what he was talking about.
But then again, this wasn't any old math hack. This was Pierre de Fermat, one of the greatest minds we've ever had on this planet. Isaac Newton himself said that Fermat's work was instrumental in forming his own ideas that led to the invention of The Calculus. When Newton said "If I have seen further it is only by standing on the shoulders of giants", he was basically talking about Kepler and Fermat. Pierre had serious street cred.
And so most mathematicians believed that Fermat did indeed find a solution to his theorem of insolubility that became known as "Fermat's Last Theorem".
And they began trying to prove it. Eventually, the solution to this problem became the greatest mystery in mathematics, because they knew a solution was possible, but the devil is in the details and an answer eluded the best minds in math for over 300 years.
Interestingly, the efforts to prove this theorem led to some amazing side discoveries in mathematics including major contributions to 19th-century Algebraic Number Theory and the 20th-century Modularity Theorem. This was one amazingly tough nut to crack.
But crack it did. In 1995, 330 years after Fermat’s death, British mathematician Andrew Wiles felled this beast with an elegant and incredibly complex 150-page proof. Margin, my ass...this was a book!
Wiles had come close in 1993, and the solution was even reported around the world, but it was later discovered that he had made an error "so abstract that it can't really be described in simple terms. Even explaining it to a mathematician would require the mathematician to spend two or three months studying that part of the manuscript in great detail." But he managed to "repair the proof" with the help of renowned Cambridge mathematician Richard Taylor, and in 1995 he had it nailed. His 1995 version of the proof for Fermat's Last Theorem stood up to the scrutiny of the world's mathematical experts. He had done it.
Wiles spent most of his career working on this proof, and he is now in the enviable position of being one of the very few mathematicians in history to solve such an intractable problem.
At last, Fermat's Last Theorem was laid to rest. Or was it?
You see, there is a little nagging thorn in the side of this theorem that just can't be pulled out. Fact is, Andrew Wiles solved this proof with 20th century mathematical concepts like the "Taniyama–Shimura Conjecture for Elliptic Curves" that simply weren't available in the 17th century in any form. In fact, most of the mathematical underpinnings of Wiles' proof were researched and discovered just to solve this problem. There is simply no chance that this is the same proof that Fermat was referring to.
So the real Fermat's Last Theorem may still be out there. Or, Fermat might just have been wrong about his proof. It is possible that we'll never know, but I will bet you that there are young mathematicians, at this very moment, trying to solve Fermat's Last Theorem with 17th century math just to answer that question.
One of the greatest mathematical minds in history left us on this day in 1665. But Pierre de Fermat wasn't quite done with us yet. You see, an innocent little inscription he left behind in one of his notebooks proceeded to baffle and challenge the best minds in mathematics for more than 300 years. Here it is:
"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain."
The ancient Pythagorean Theorem says that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. In easier to understand algebra, that looks like this:
x^2 + y^2 = z^2
And example would be: 5^2 + 12^2 = 13^2, but there are many more.
Know it or not, we use this every day. It's one of the most fundamental and important relationships in all of geometry. We couldn't design a thing without it. Even grade school kids learn this.
But what Fermat was saying is that for any power of this equation higher than 2, there is no whole number solution. So don't look for it. An example would be:
x^5 + y^5 = z^5
According to Fermat, this has no whole number solution, and it never will. And he can prove it. In fact he DID prove it. He just ran out of room in the margins of his notebook and so he left it to us to re-discover.
Now, I don't know about you, but if someone told me they had solved a very difficult problem but couldn't show you because they ran out of paper, I might be skeptical. And many people were. They said he was either messing with us, or he didn't know what he was talking about.
But then again, this wasn't any old math hack. This was Pierre de Fermat, one of the greatest minds we've ever had on this planet. Isaac Newton himself said that Fermat's work was instrumental in forming his own ideas that led to the invention of The Calculus. When Newton said "If I have seen further it is only by standing on the shoulders of giants", he was basically talking about Kepler and Fermat. Pierre had serious street cred.
And so most mathematicians believed that Fermat did indeed find a solution to his theorem of insolubility that became known as "Fermat's Last Theorem".
And they began trying to prove it. Eventually, the solution to this problem became the greatest mystery in mathematics, because they knew a solution was possible, but the devil is in the details and an answer eluded the best minds in math for over 300 years.
Interestingly, the efforts to prove this theorem led to some amazing side discoveries in mathematics including major contributions to 19th-century Algebraic Number Theory and the 20th-century Modularity Theorem. This was one amazingly tough nut to crack.
But crack it did. In 1995, 330 years after Fermat’s death, British mathematician Andrew Wiles felled this beast with an elegant and incredibly complex 150-page proof. Margin, my ass...this was a book!
Wiles had come close in 1993, and the solution was even reported around the world, but it was later discovered that he had made an error "so abstract that it can't really be described in simple terms. Even explaining it to a mathematician would require the mathematician to spend two or three months studying that part of the manuscript in great detail." But he managed to "repair the proof" with the help of renowned Cambridge mathematician Richard Taylor, and in 1995 he had it nailed. His 1995 version of the proof for Fermat's Last Theorem stood up to the scrutiny of the world's mathematical experts. He had done it.
Wiles spent most of his career working on this proof, and he is now in the enviable position of being one of the very few mathematicians in history to solve such an intractable problem.
At last, Fermat's Last Theorem was laid to rest. Or was it?
You see, there is a little nagging thorn in the side of this theorem that just can't be pulled out. Fact is, Andrew Wiles solved this proof with 20th century mathematical concepts like the "Taniyama–Shimura Conjecture for Elliptic Curves" that simply weren't available in the 17th century in any form. In fact, most of the mathematical underpinnings of Wiles' proof were researched and discovered just to solve this problem. There is simply no chance that this is the same proof that Fermat was referring to.
So the real Fermat's Last Theorem may still be out there. Or, Fermat might just have been wrong about his proof. It is possible that we'll never know, but I will bet you that there are young mathematicians, at this very moment, trying to solve Fermat's Last Theorem with 17th century math just to answer that question.
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