How can I reduce the time to solve this equation with Integral solutions?

Question

I know that $\sqrt[3]{2744} = 2 + 7 + 4 + 4 - 3$, $\sqrt[3]{3375} = 3 + 3 + 7 + 5 - 3$, and $\sqrt[3]{4096} = 4 + 0 + 9 + 6 - 3$ by

 Solve[{a + b + c + d + e - 3 == CubeRoot[10000 a + 1000 b + 100 c + 10 d + e], 1 <= a <= 9, 0 <= b <= 9, 0 <= c <= 9, 0 <= d <= 9, 0 <= e <= 9}, {a, b, c, d, e}, Integers]

I am trying to solve

Solve[{a + b + c + d + e - 3 == 
   CubeRoot[10000 a + 1000 b + 100 c + 10 d + e], 1 <= a <= 9, 
  0 <= b <= 9, 0 <= c <= 9, 0 <= d <= 9, 0 <= e <= 9}, {a, b, c, d, 
  e}, Integers]

But the time run too long and I can not get the result. How can I reduce the time to solve this equation?

Answer 1

Try

x=10^4;
While[(Total[IntegerDigits[x]]-3)^3!=x&&x<10^5,x++];
If[x==10^5,"No solution",x]

which finishes in a second and displays No solution

But there is a solution for (Total[IntegerDigits[x]]+3)^3

Answer 2

The fastest way: Use Table!

Flatten[Table[
   If[(a + b + c + d + e - 3)^3 == 
     10000 a + 1000 b + 100 c + 10 d + e, {a, b, c, d, e}, 
    Nothing], {a, 0, 9}, {b, 0, 9}, {c, 0, 9}, {d, 0, 9}, {e, 0, 
    9}] , 5 - 1  ] // AbsoluteTiming
(*{0.34782, {{0, 0, 1, 2, 5}, {0, 0, 2, 1, 6}, {0, 0, 3, 4, 3}, {0,2, 7, 4, 4}, {0, 3, 3, 7, 5}, {0, 4, 0, 9, 6}}}*)

Adapting @Bill's interesting answer we get the short problemdescription

Table[If[(Total[IntegerDigits[x]] - 3)^3 == x, x,Nothing], {x, 10^5}]
(*{125, 216, 343, 2744, 3375, 4096}*)

Answer 3

Reduce[{(a + b + c + d + e - 3)^3 == 
   a*10^4 + b*10^3 + c*10^2 + d*10 + e, 
  0 <= {a, b, c, d, e} <= 9}, {a, b, c, d, e}, Integers]

(a == 0 && b == 0 && c == 1 && d == 2 && e == 5) || (a == 0 && 
   b == 0 && c == 2 && d == 1 && e == 6) || (a == 0 && b == 0 && 
   c == 3 && d == 4 && e == 3) || (a == 0 && b == 2 && c == 7 && 
   d == 4 && e == 4) || (a == 0 && b == 3 && c == 3 && d == 7 && 
   e == 5) || (a == 0 && b == 4 && c == 0 && d == 9 && e == 6)

It means that there no solution for a!=0. The same as

Solve[{(a + b + c + d + e - 3)^3 == 
   a*10^4 + b*10^3 + c*10^2 + d*10 + e, 
  0 <= {a, b, c, d, e} <= 9}, {a, b, c, d, e}, Integers]

and

Reduce[{(a + b + c + d + e - 3)^3 == 
   a*10^4 + b*10^3 + c*10^2 + d*10 + e, 0 <= {a, b, c, d, e} <= 9, 
  a != 0}, {a, b, c, d, e}, Integers]

False

and there three solutions for the first case.

Solve[{(a + b + c + d - 3)^3 == a*10^3 + b*10^2 + c*10 + d, 
  0 <= {a, b, c, d} <= 9, a != 0}, {a, b, c, d}, Integers]

{{a -> 2, b -> 7, c -> 4, d -> 4}, {a -> 3, b -> 3, c -> 7, d -> 5}, {a -> 4, b -> 0, c -> 9, d -> 6}}