I keep losing a variable

[procyon]

Hi,

I started with this (one of several) equation

\(\displaystyle b^2+c^2=2(4u^4+v^4 )^2\)

for which I can merrily find a general diophantine solution with \(\displaystyle r\) and \(\displaystyle s\) such that

\(\displaystyle 2u^2=4r^4-2s^4\)

\(\displaystyle v^2=4r^2s^2\)

\(\displaystyle b=16r^8+16r^4 s^4-4s^8\)

and \(\displaystyle c=16r^8-16r^4 s^4-4s^8\)

(These could have been scaled down, but I also require access to \(\displaystyle u^2\) so this is handier)

In the other equations I also need access to \(\displaystyle v\) and \(\displaystyle u\)

\(\displaystyle v\) isn't a problem as, from the above, \(\displaystyle v=2rs\)

The trouble comes with \(\displaystyle u\)

As you'll see, at the moment I have \(\displaystyle u^2=2r^4-s^4\)

Any method I use to get a general solution to \(\displaystyle u^2=2r^4-s^4\) obsoletes \(\displaystyle r\) and \(\displaystyle s\) in the same manner, and these are required values for \(\displaystyle v\)

Any ideas how I can solve \(\displaystyle u^2=2r^4-s^4\) for integer \(\displaystyle u\) and still keep integer values for \(\displaystyle r\) and \(\displaystyle s\)?

Thanks

 

[lookagain]

Any ideas how I can solve \(\displaystyle u^2=2r^4-s^4\) for integer \(\displaystyle u\) and still keep integer values for \(\displaystyle r\) and \(\displaystyle s\)?

Given that r and s must belong to the set of integers:


\(\displaystyle When \ |s| \ = \ |r|, \ the \ equation \ becomes \)

\(\displaystyle \ u^2 \ = \ 2r^4 - r^4, \ which \ becomes\)

\(\displaystyle u^2 \ = \ r^4\)

r and s could equal any integers, as long as their absolute values equal each other.

\(\displaystyle Then \ u \ = \ \pm r^2.\)



----> I am not sure if I addressed your intended question. <-----

 

[procyon]

Thanks lookagain

My mistake, I should have specified that none of the |values| can be equal

I am looking for all values produced to be \(\displaystyle +ve\) integers and setting \(\displaystyle |r|=|s|\)

leaves \(\displaystyle c=16r^8-16r^4s^4-4s^8\) as \(\displaystyle c=-4r^8\)

Pro