Determine if $x^3+y^3+z^3+t^3 = 10^{2021}$ has a solution

Question

I want to know if the equation $x^3+y^3+z^3+t^3=10^{2021}$ has distinct positive integer solutions PowersRepresentations[10^2021, 4, 3]
return

PowersRepresentations::ovfl: Overflow occurred in computation.

FindInstance[{x^3 + y^3 + z^3 + t^3 == 10^2021, 0 < x < y < z < t}, {x,y,z,t}, Integers]

My computer runs too long. How can I reduce timing to solve this equation?

Answer 1

Easy, notice that $10^{2021}=100\times 10^{3\times 673}$. Next use your code, but for the factor 100.

FindInstance[{x^3 + y^3 + z^3 + t^3 == 100, 0<x<y<z<t}, {x,y,z,t}, Integers]

yielding a single result

(*{{x -> 1, y -> 2, z -> 3, t -> 4}}*)

Now verify the solution

(x^3 + y^3 + z^3 + t^3 /. {x -> 1 10^673,y -> 2 10^673,z -> 3 10^673,t -> 4 10^673}) == 10^2021
(* True*)

Answer 2

Two solutions may be found starting from: $$10^{2021}=10^5\times10^{2016}=12500\times2^3\times10^{3\times672}$$

Since $12500=19^3+17^3+8^3+6^3=18^3+17^3+12^3+3^3$ we have:

$$10^{2021}=(38\times10^{672})^3+(34\times10^{672})^3+(16\times10^{672})^3+(12\times10^{672})^3$$ $$10^{2021}=(36\times10^{672})^3+(34\times10^{672})^3+(24\times10^{672})^3+(6\times10^{672})^3$$