What are diophantine equations?

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August 6th 2008, 06:02 AM
fardeen_gen

What are diophantine equations?

What are diophantine equations? And what is Fermat's method of infinite descent? Can anyone provide me some good resources on the web for learning them in detail?

EDIT. Can someone on MHF start a tutorial on Number theory? It would be very nice.
August 6th 2008, 10:16 AM
ThePerfectHacker

Quote:

Originally Posted by fardeen_gen

What are diophantine equations?

It is an equation where the solution is in terms of positive integers.
For example, solve $x^3+y^3=z^3$ .
It turns out this equation has no solution - proving that is a lot more complicated.

Quote:

And what is Fermat's method of infinite descent?

It is a method of arriving at a contradiction. Say we want to show the equation above has no solution. Thus, we assume that there exists a solution, for example some $z$ which makes that equation solvable. And then having that assumption we show a smaller solution $z_0$ can be constructed from the one we had. This leads to a contradiction. Because if we repeat the argument we can obtain even a smaller solution from $z_0$ . And we can do this indefinitely. Since the positive integers cannot be indefinitely decreased our initial assumption must have been wrong.

Quote:

EDIT. Can someone on MHF start a tutorial on Number theory? It would be very nice.

That takes too much time. It is almost as bad as asking a person if we can write a book on some topic. What do you think the response will be?
August 6th 2008, 02:39 PM
Moo

Hello,

Quote:

Originally Posted by fardeen_gen

What are diophantine equations?

Basically, it's an equation of this form : P(x,y,z)=0.
P is a polynomial with x,y,z as variables and such that the coefficients are integers or rational numbers.
The solutions are also searched in the range of integers or rational numbers.

Talkin' about Fermat, the example TPH provided is an element of Fermat's last theorem : $x^n+y^n=z^n$ has no integer solution for $n>2$ (Tongueout)