Difference between revisions of "Fermat's Last Theorem"

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'''Fermat's Last Theorem''' is a recently proven [[theorem]] stating that for positive [[integers]] <math>a,b,c,n</math> with <math>n \geq 3</math>, there are no solutions to the equation <math>a^n + b^n = c^n</math>.
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==History==
 
==History==
Fermat's last theorem is particularly famous in number theory.  It was proposed by Fermat in the margin of his book Arithmetica.  The note in the margin (when translated) read: "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."  However, nobody was able to prove this theorem for centuries.  Finally, in 1993 Andrew Wiles produced a proof of the theorem - a proof that was much more complicated than anything Fermat could have produced himself.
 
  
==The Theorem==
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Fermat's Last Theorem was proposed by [[Pierre Fermat]] in the <math>1600s</math> in the margin of his copy of the book ''Arithmetica''.  The note in the margin (when translated) read: "It is impossible for a [[perfect cube | cube]] to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. ''I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.''"
  
Given integers <math>a,b,c</math> and <math>n</math>, with <math>n \geq 3</math> there are no solutions to the equation:
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Many mathematicians today doubt that Fermat actually had a proof for this theorem.  If he did have one, he never published it, though he did publish a proof for the case <math>n=4</math>.  It seems unlikely that he would have circulated a proof for the special case when he had a general solution. Some think that Fermat's proof was flawed, and that he saw the flaw after a time.
  
<math>\displaystyle a^n + b^n = c^n</math>
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Some mathematicians have suggested that Fermat had a proof that relied on unique factorization in [[ring]]s of the form <math>\mathbb{Z}[\sqrt[n]{-1}]</math>.  Unfortunately, this is not often the case.  In fact, it has now been known for some time how to solve the problem when this is the case.
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Despite Fermat's claim that a simple proof existed, the theorem wasn't proven until [[Andrew Wiles]] did so in <math>1993</math>.  Wiles's proof was the culmination of decades of work in number theory.  Interestingly enough, Wiles's proof was much more modern than anything Fermat could have produced himself.  It exploited connections between modular forms and elliptic curves.
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In some sense, Fermat's last theorem is a dead end: it has led to few new mathematical consequences.  However, the search for the proof of the theorem generated whole new areas of mathematics.  In this sense, it was a good, productive problem.
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The ABC Conjecture is a far-reaching conjecture that implies Fermat's Last Theorem for <math>n \ge 7</math>.  It is one of the most famous still-open problems in number theory, although a proof has been announced and is being verified (current as of August <math>2013</math>).
  
 
==See Also==
 
==See Also==
 
* [[Number Theory]]
 
* [[Number Theory]]
** [[Diophantine Equations]]
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** [[Diophantine equation]]s
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* [[Andrew Wiles]]
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[[Category:Number theory]]
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[[Category:Theorems]]

Latest revision as of 19:45, 11 June 2018

Fermat's Last Theorem is a recently proven theorem stating that for positive integers $a,b,c,n$ with $n \geq 3$, there are no solutions to the equation $a^n + b^n = c^n$.

History

Fermat's Last Theorem was proposed by Pierre Fermat in the $1600s$ in the margin of his copy of the book Arithmetica. The note in the margin (when translated) read: "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."

Many mathematicians today doubt that Fermat actually had a proof for this theorem. If he did have one, he never published it, though he did publish a proof for the case $n=4$. It seems unlikely that he would have circulated a proof for the special case when he had a general solution. Some think that Fermat's proof was flawed, and that he saw the flaw after a time.

Some mathematicians have suggested that Fermat had a proof that relied on unique factorization in rings of the form $\mathbb{Z}[\sqrt[n]{-1}]$. Unfortunately, this is not often the case. In fact, it has now been known for some time how to solve the problem when this is the case.

Despite Fermat's claim that a simple proof existed, the theorem wasn't proven until Andrew Wiles did so in $1993$. Wiles's proof was the culmination of decades of work in number theory. Interestingly enough, Wiles's proof was much more modern than anything Fermat could have produced himself. It exploited connections between modular forms and elliptic curves.

In some sense, Fermat's last theorem is a dead end: it has led to few new mathematical consequences. However, the search for the proof of the theorem generated whole new areas of mathematics. In this sense, it was a good, productive problem.

The ABC Conjecture is a far-reaching conjecture that implies Fermat's Last Theorem for $n \ge 7$. It is one of the most famous still-open problems in number theory, although a proof has been announced and is being verified (current as of August $2013$).

See Also

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