Solve 1729 as Sum of 2 Cubes: Natural Numbers

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Homework Help Overview

The problem involves finding two different representations of the number 1729 as the sum of two cubes using natural numbers. Participants are exploring the mathematical properties and relationships involved in this representation.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to find solutions through guess and check but expresses uncertainty about analytical methods. Some participants suggest using algebraic identities to simplify the problem, while others recall methods involving complex numbers and moduli.

Discussion Status

The discussion is ongoing, with participants sharing different approaches and recalling methods from previous learning experiences. There is no explicit consensus, but various lines of reasoning are being explored.

Contextual Notes

Participants are working under the constraint that the solutions must involve natural numbers, and there is a reference to a well-known anecdote related to the number 1729.

cragar
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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.
 
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You can simplify a little by using [itex]x^3+ y^3= (x+ y)(x^2- xy+ y^2)[/itex].

Now, factor 1729 to find two factors that would fit that.
 
sweet thanks for the help
 
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?
 
You can't not know the oft-told tale about this question and this number?
 
There is a very oft-told tale about this question with this number many people here will know.
 
Last edited:

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