<s>[INST] Determine if $x^3+y^3+z^3+t^3 = 10^{2021}$ has a solution 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?
 [/INST] 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:
$$10^{2021}=(38\times10^{672})^3+(34\times10^{672})^3+(16\times10^{672})^3+(12\times10^{672})^3$$
$$10^{2021}=(36\times10^{672})^3+(34\times10^{672})^3+(24\times10^{672})^3+(6\times10^{672})^3$$
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