432 Is Special

Let S S be the set of rational numbers x x such that x 3 432 x^3-432 is the square of a rational number.

If S S is infinite, enter 1 -1 . If S S is finite, enter the sum of the elements of S . S.

Here is a very helpful hint:
If y 2 = x 3 432 , y^2=x^3-432, let u = 36 y 6 x , v = 36 + y 6 x u = \dfrac{36-y}{6x}, \, v = \dfrac{36+y}{6x} . What is u 3 + v 3 u^3+v^3 ?


The answer is 12.

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1 solution

Michael Mendrin
Mar 28, 2016

Using the very helpful hint (thanks, Patrick!), let

u = 36 y 6 x , v = 36 + y 6 x u = \dfrac{36-y}{6x}, \, v = \dfrac{36+y}{6x}

Then

u 3 + v 3 = 432 + y 2 x 3 u^3+v^3=\dfrac{432+{y}^{2}}{{x}^{3}}

so that we immediately have, as a consequence

u 3 + v 3 = 1 u^3+v^3=1

which has no known rational solutions per Fermat's Last Theorem , other than

( u , v ) = ( 1 , 0 ) (u,v)=(1,0) or ( u , v ) = ( 0 , 1 ) (u,v)=(0,1)

which leads to the only rational solutions

( x , y ) = ( 12 , 36 ) (x,y)=(12,-36) or ( x , y ) = ( 12 , 36 ) (x,y)=(12,36)

and so the sum of the elements of S S is just 12 12

@Patrick Corn , can we always find a substitution of u u and v v (as rational functions of y y and x x ), where y 2 = x 3 r y^2=x^3- r , for another rational r r ?

E.g.: I don't know how to formulate u u and v v for y 2 = x 3 11 y^2=x^3 - 11 . And yes, I was thinking about solving this problem by using your hint here.

Pi Han Goh - 3 years, 5 months ago

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No, pretty much only for this specific r r (up to multiplication by a perfect sixth power). The family of elliptic curves y 2 = x 3 D y^2 = x^3 - D is much, much more interesting than the one Fermat curve u 3 + v 3 = 1. u^3+v^3=1. Many such curves have lots of points--in fact, there are lots of open questions about the structure of those points, including one of the Millennium prize problems .

I highly recommend Joe Silverman's book The Arithmetic of Elliptic Curves . Lots of down-to-earth examples, and a good overview of the theory. (He also has a book for undergraduates called Rational Points on Elliptic Curves , which I don't know as much about but I assume is also good and probably significantly easier.)

Patrick Corn - 3 years, 5 months ago

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Thanks for your detailed response. I'll try to get the book as well. =D

Pi Han Goh - 3 years, 5 months ago

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