Solve 1729 as Sum of 2 Cubes: Natural Numbers

  • Thread starter Thread starter cragar
  • Start date Start date
  • Tags Tags
    Sum
AI Thread Summary
1729 is the smallest positive integer expressible as the sum of two cubes in two distinct ways using natural numbers. The equation x^3 + y^3 = 1729 can be approached analytically by factoring it as (x + y)(x^2 - xy + y^2). Participants in the discussion suggest using guess and check methods, while also recalling techniques involving complex numbers and moduli for similar problems. The conversation highlights the historical significance of 1729, often referred to in mathematical anecdotes. Ultimately, the focus remains on finding the two specific pairs of natural numbers that satisfy the equation.
cragar
Messages
2,546
Reaction score
3

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.
 
Physics news on Phys.org
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.
 
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:

Thread 'Finding the number of ways to arrange identical balls in a circle (3 different colors)'

I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
View full post »

Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
View full post »

Similar threads

Write 1729 as the sum of two cubes
Replies
10
Views
3K
How do I approach this problem? (Cubes within a larger cube....)
Replies
21
Views
4K
MHB Sum of Numbers on Cube Faces to Equal 2004
Replies
1
Views
2K
Number of ways of arranging 7 characters in 7 spaces
Replies
8
Views
2K
How to simplify cube root expression
Replies
5
Views
2K
How to Calculate the Edge Length of a Cube Given Its Volume and Surface Area?
Replies
10
Views
2K
A+b=236, What numbers to get a maximum product ab?
Replies
21
Views
3K
Express this sum as a fraction of whole numbers
Replies
14
Views
3K
Is a^2 Always Positive for a Real Number a?
Replies
18
Views
3K
Finding number of natural numbers N such that............
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top