Diophantine euations

[computerages]

hi every1~!

 

I was just curious that what is the point of studying only integar solutions of a problem. One reason that comes up to my mind is that some irrational numbers have infinite numbers after decimal, so therefore it would be a problem to write an exact solution. But are there any more reasons that prevent us playing with other numbers?

[matt grime]

Irrationals do not have infinite numbers after their decimal places (there is no such thing as an infinite number in a decimal representation system). They *all* have non-repeating decimal expansions, however. There is no problem writing an exact solution to any equation, by the way. It is just your preference for thinking that 2 is exact but sqrt(2) is not.

 

The point is that rational numbers, and integers are more natural than the real numbers, they are what has fascinated mankind throughout the centuries.

[Dave]

More than that, the study of the solution of Diophantine equations pretty much inspired the creation of modern day algebra - rings of integers, ideal classes, stuff like that.

 

For example, an equation like [imath]x^2 + 5 = y^3[/imath] is not easy to solve by the classical methods of say, reduction modulo n or infinite descent. However, if we apply some basic algebra theory - in this case, the theory of prime ideals - then it's easy to show that there are only finitely many solutions and determine what those solutions are.