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| + | Fermat’s Last Theorem | ||
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| + | Fermat’s Last Theorem states that for the Diophantine equation x^n + y^n = z^n where x, y, and z are integers has no solutions that are nonzero when n>2. | ||
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| + | Background: Around 1637 Fermat claimed to have proven this theorem. He wrote in the margins of one of his books that when n>2 there were no solutions to the Diophantine equation. However, he did not prove his claim. 358 years later Andrew Wiles was able to successfully prove Fermat’s last theorem. | ||
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| + | An example of a Proof of n=3: | ||
| + | x^3 + y^3 = z^3 | ||
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| + | x^3 -1=(x-1)(x-t)(x-t^2) | ||
| + | where t=e^2πi/3 | ||
| + | Let x=x/y | ||
| + | Can be rewritten as (x^3)/(y^3) – 1=(x/y -1)(x/y –t)(x/y –t^2) | ||
| + | Now multiply both sides b y^3 and y become –y | ||
| + | Can be rewritten as | ||
| + | z^3 = x^3 + y^3 = (x+y)(x+ty)(x+(t^2)y) | ||
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| + | By looking at the prime factors of the left side of the above equation, it is impossible for it to equal z^3. | ||
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| + | Andrew Wiles came up with an extensive proof for all “n” in 1994 and it was later published in 1995. | ||
| + | |||
| + | Resources: | ||
Revision as of 13:00, 26 April 2014
Any integer factors, more or less uniquely, into prime numbers. Does this question make sense for real numbers? What about numbers of the form a+b*i where i^2=-1 and a,b are integers? What about numbers of the form a+b*root(N) where a,b,N are integers?
(Possible appendix: What does this concept have to do with Fermat's Last theorem, and what does FLT say anyway?)
Fermat’s Last Theorem
Fermat’s Last Theorem states that for the Diophantine equation x^n + y^n = z^n where x, y, and z are integers has no solutions that are nonzero when n>2.
Background: Around 1637 Fermat claimed to have proven this theorem. He wrote in the margins of one of his books that when n>2 there were no solutions to the Diophantine equation. However, he did not prove his claim. 358 years later Andrew Wiles was able to successfully prove Fermat’s last theorem.
An example of a Proof of n=3: x^3 + y^3 = z^3
x^3 -1=(x-1)(x-t)(x-t^2) where t=e^2πi/3 Let x=x/y Can be rewritten as (x^3)/(y^3) – 1=(x/y -1)(x/y –t)(x/y –t^2) Now multiply both sides b y^3 and y become –y Can be rewritten as z^3 = x^3 + y^3 = (x+y)(x+ty)(x+(t^2)y)
By looking at the prime factors of the left side of the above equation, it is impossible for it to equal z^3.
Andrew Wiles came up with an extensive proof for all “n” in 1994 and it was later published in 1995.
Resources:
