Vampire Count Kim Sang-geun

Time limit2sMemory limit128 MB

Summary

For each n under 10000, find natural a, b, c with a >= c, a^3 + c^3 = n*b^3, minimizing a + b + c + b, or print No value if the minimum exceeds 4000.

Level: Medium7 of 10
Topics: Number theory, Math, Brute force
Solved: No attempts yet

Problem

To fend off the attacks of the vampire Count Kim Sang-geun, you must memorize an equation of the following form.

$\left(\frac{a}{b}\right)^3 + \left(\frac{c}{b}\right)^3 = n$

Here $a$ , $b$ , $c$ , and $n$ are all natural numbers. For example, when $n = 9$ the following equation holds.

(415280564497/348671682660)^3 + (676702467503/348671682660)^3 = 9

But an equation this long cannot be memorized before the Count strikes and turns you into a vampire. Fortunately, for the same $n = 9$ there is a far shorter equation.

(2/1)^3 + (1/1)^3 = 9

Given a natural number $n$ , write a program that finds the equation that is easiest to memorize. The easiest-to-memorize equation is the one that minimizes $a + b + c + b$ , subject to $\frac{a}{b} \ge \frac{c}{b}$ (that is, $a \ge c$ ). The equation satisfying these conditions is always unique. If no equation has $a + b + c + b$ smaller than $4000$ , print No value..

Input

The input consists of several test cases. Each line contains one natural number $n$ smaller than $10000$ . The last line contains a single $0$ , which is not processed.

Output

For each test case, print the easiest-to-memorize equation on its own line. Print exactly one space before and after each + and = sign. If no valid equation exists, print No value..

Examples5

Example 1

Input

Expected output

No value.
(2/1)^3 + (1/1)^3 = 9
(5/3)^3 + (4/3)^3 = 7
(370/21)^3 + (170/21)^3 = 6000

Example 2
Input
```
2
0
```
Expected output
```
(1/1)^3 + (1/1)^3 = 2
```
Example 3
Input
```
16
0
```
Expected output
```
(2/1)^3 + (2/1)^3 = 16
```

Example 4

Input

6
13
0

Expected output

(37/21)^3 + (17/21)^3 = 6
(7/3)^3 + (2/3)^3 = 13

Example 5

Input

Expected output

(5/2)^3 + (3/2)^3 = 19
(19/7)^3 + (1/7)^3 = 20
(89/39)^3 + (19/39)^3 = 12