How can I find the general solution of $a^4+b^4=c^4+d^4$ ($a,b,c,d > 0$)?
And how did Euler find the solution $158^4+59^4=133^4+134^4$?



Diophantine equation - $a^4+b^4=c^4+d^4$ ($a,b,c,d > 0$)
Euler wrote $a^4-d^4=c^4-b^4$ with $a=p+q, d=p−q, c=r+s$ and $b=r−s$, obtaining
$pq(pp + qq) = rs(rr + ss)$, and then did several other special transformations, until
he arrived at the special solution
$$
a=p+q=2219449, c=r+s=1584749, 
$$


$$
b=r−s=−555617, d=p−q=−2061283
$$
which satisfies $a^4 + b^4 = c^4+d^4$.  

All the details can be found here: <http://arxiv.org/pdf/math/0505629v2.pdf>.


