From: fredh@ix.netcom.com (Fred W. Helenius)
Subject: Re: Problem related to FLT
Date: Fri, 25 Jun 1999 21:45:33 GMT
Newsgroups: sci.math
Keywords: two cubes almost sum to another

mathwft@math.canterbury.ac.nz (Bill Taylor) wrote:

>Bob Silverman <bobs@rsa.com> writes:
>
>|> For example, it is known that there are infinitely many (x,y,z)
>|> such that x^3 + y^3 = z^3 + 1
>
>Is there some simple family of solutions?   

There is.  Hardy & Wright (Theorem 235) give a parametric form for
nontrivial rational solutions of x^3 + y^3 = u^3 + v^3 as follows:

x = r(1 - (a - 3b)(a^2 + 3b^2))
y = r((a + 3b)(a^2 + 3b^2) - 1)
u = r((a + 3b) - (a^2 + 3b^2)^2)
v = r((a^2 + 3b^2)^2 - (a - 3b))

where r,a,b are rational and r <> 0.

Letting r = 1 and a = 3b, we obtain

x = 1
y = 72b^3 - 1
u = 6b - 144b^4
v = 144b^4

which are integers whenever b is.  (Positive integer solutions can be
obtained by using negative values of b and moving the two negative
cubes across the equal sign.)

Unfortunately, not every integer solution derives from integer values
of r,a,b.  The best known, 1^3 + 12^3 = 9^3 + 10^3, corresponds to
r = -361/42, a = 10/19, b = -7/19.

--
Fred W. Helenius	<fredh@ix.netcom.com>
==============================================================================

From: pete@bignode.southern.co.nz (Pete Moore)
Subject: Re: Problem related to FLT
Date: 29 Jun 99 10:47:25 +1200
Newsgroups: sci.math

Bill Taylor (mathwft@math.canterbury.ac.nz) wrote:
>Bob Silverman <bobs@rsa.com> writes:
>
>|> > x=z, y=1 ?     Looks simple to me.
>|> 
>|> Typical inane newsgroup comment.
>
>Wasn't it though!    :)
>
>
>|> There is an infinite-family  parameterization  of rational solutions
>|> 
>|> to x^3 + y^3 = a^3 + b^3.
>
>Good.   So can you tell us what it is, please!

Apart from the trivial solutions  {x=y=0,a=-b} and {x=a,y=b}, the general
solution in rationals is 

x = u(1-(v-3w)(v^2+3w^2))
y = u((v+3w)(v^2+3w^2)-1)
a = u((v+3w)-(v^2+3w^2)^2)
b = u((v^2+3w^2)^2-(v-3w))

where u,v,w are any rationals (provided u<>0).

(From Hardy & Wright, section 13.7).

-- 
+----------------  pete@bignode.southern.co.nz  ----------------+
 The effort to understand the universe is one of the very few things
 that lifts human life above the level of farce, and gives it some
 of the grace of tragedy   -  Steven Weinberg