Is $524154366113525716400386$ the sum of two fourth-powers? I suspect that this number is the sum of two fourth powers. Can anyone use a computer program, or SAGE, or Wolfram Alpha to check whether this number is the sum of two fourth- powers?
The complete factorization of this number is given by : $$524154 366113 525716 400386 = 2 × 521 × 8761 × 21529 × 221281 × 12 052297$$ Notice that all the factors are of the form $16n+1$ or $16n+9$ (They are all congruent to $1$ or $9 \mod16$ ). Hence this number may be the sum of two-fourth powers.
In case you were not aware, Mathworld mentions Elkies' disproof, citing On $A^4+B^4+C^4=D^4$.
Without having any software installed on your computer, you can check that this particular number is not the sum of two fourth powers by, say, pasting
FindInstance[(524154366113525716400386524154366113525716400386==x^2+y^2)~And~(x==z^2)~And~(y==w^2),{x,y,z,w},Integers]into the wolfram cloud sandbox and use Shift+Enter or numpad Enter or clicking on Gear>"Evaluate Cell" to run it. (For confirmation that this code would work, compare to using280286069726155265499093303843106.)There may be some modular congruences that rule this number out as well.