TITLE: Explain these integer solutions
QUESTION [5 upvotes]: $a+b=u^2, a^2+b^2=v^4$
I have found the solution:
$a=4565486027761, \quad b=1061652293520, \quad u=2372159, \quad v=2165017$.
But I do not know a more theoretical way to get them.

REPLY [2 votes]: Regarding above equation's shown below:
$a+b=u^2$  and
$a^2+b^2=v^4$
Solution is shown below:
$(a,b,u,v)=[(120), (-119), (1), (13)]$
There is another solution given on Seiji Tomita site and the link is shown below.
        http://www.maroon.dti.ne.jp/fermat

And click on computational number theory and then click on article # 146
The numerical solution is shown below:
$a=214038981475081188634947041892245670988588201$ 
$b=109945628264924023237017010068507003594693720$
$u= 17999572487701067948161$
$v=15512114571284835412957$