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

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?

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

• If you start with x=1 your code finds the solution 125, But your code has to be restarted with last x-value to find all solution. Commented Sep 2, 2022 at 8:00

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}}}*)


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

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}}