Difference between revisions of "Fermat's Last Theorem"
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| − | '''Fermat's Last Theorem''' is a | + | '''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>. |
==History== | ==History== | ||
| − | |||
| − | == | + | Fermat's last theorem was proposed by [[Pierre Fermat]] in the 1600s in the margin of his copy of the book ''Arithmetica'', by [[Diophantus]]. 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." |
| + | |||
| + | 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 | ||
| + | circulate a proof for the case <math>n=4</math>. It seems unlikely that he would | ||
| + | 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 [[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. | ||
| + | |||
| + | 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 form]]s and [[elliptic curve]]s. | ||
| + | |||
| + | 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 <math>n \ge 7</math>. It is one of the most famous | ||
| + | still-open problems in number theory. | ||
| + | |||
| + | == Resources == | ||
| + | |||
* [http://www.amazon.com/exec/obidos/ASIN/0385493622/artofproblems-20 Fermat's Enigma] | * [http://www.amazon.com/exec/obidos/ASIN/0385493622/artofproblems-20 Fermat's Enigma] | ||
| + | * [http://www.youtube.com/watch?v=qiGOxGEbaik Documentary on Fermat's Last Theorem] | ||
==See Also== | ==See Also== | ||
Revision as of 00:45, 10 April 2009
Fermat's Last Theorem is a recently proven theorem stating that for positive integers 
with 
, there are no solutions to the equation 
.
History
Fermat's last theorem was proposed by Pierre Fermat in the 1600s in the margin of his copy of the book Arithmetica, by Diophantus. 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
circulate a proof for the case 
. It seems unlikely that he would
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
![$\mathbb{Z}[\sqrt[n]{-1}]$](http://latex.artofproblemsolving.com/4/5/3/45382108f0b8b08fde3ff55f436a0255f25dca4e.png)
. 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 
. It is one of the most famous
still-open problems in number theory.
