I am looking for two integers, $a$ and $b$, such that $a^4+b^4=n$ and where $n$ is one of the six numbers
11570282017820433013523535601, or
11570314155463181637103702801, or
11572060353961555386606814001, or
11572215695702429026631328801, or
11573624522376724598676284401, or
11575215560569326509742400801
Can anyone give me a software program to check whether an integer is the sum of two fourth powers ?
$n=11570282017820433013523535601$ satisfies $7\mid\mid n$, i.e., $7\mid n$ and $7^2\nmid n$, so that $n$ is not the sum of two squares, according to the Fermat's criterion. But then $n=a^4+b^4=(a^2)^2+(b^2)^2$ is impossible, too.
For $n=11570314155463181637103702801$ we have $2699\mid\mid n$, and $2699\equiv 3 \bmod 4$, hence also not of the form $x^2+y^2$. Similarly for thers, e.g. $n=11572215695702429026631328801$, see the comment of vrugtehagel.
However, $n=11572060353961555386606814001$ is the sum of two squares. With pari gp, sage or another CAS we can check whether it is a sum of two fourth powers, or not.
