Solve 1729 as Sum of 2 Cubes: Natural Numbers

[cragar]

Homework Statement

1729 is the smallest positive integer that can be represented in two different ways as the sum of two cubes , what are the two ways.
They have to be natural numbers.

The Attempt at a Solution

x^3+Y^3=1729 i could just find the answer by guess and check , but I am not sure how to do it analytically.

 

[HallsofIvy]

You can simplify a little by using $x^3+ y^3= (x+ y)(x^2- xy+ y^2)$.

Now, factor 1729 to find two factors that would fit that.

 

[cragar]

sweet thanks for the help

 

[Mentallic]

I remember my teacher showing us a method with complex numbers and moduli to find the sum of two squares to equal some number we have. If only I remember the method... Maybe this can be extended to the sum of two cubes?

 

[epenguin]

You can't not know the oft-told tale about this question and this number?

 

[epenguin]

There is a very oft-told tale about this question with this number many people here will know.