Problem 1241
The Gougu Theorem
Time Limit: 2000ms
Memory Limit: 65536kb
Pythagorean Theorem is "humanity's greatest scientific discoveries of the ten" is a basic elementary geometry theorems. "This theorem has a very long history, almost all ancient civilizations (Greece, China, Egypt, Babylon, India, etc.) have studied this theorem. Pythagorean Theorem in the West known as the Pythagorean Theorem, are said to Ancient Greek mathematician and philosopher Pythagoras (Pythagoras, BC572 ~ BC497) was first discovered in BC550.
Pythagorean Theorem in China known as the Gougu/勾股 Theorem. Around 1100 BC, the Western Zhou period, the ancient Chinese mathematician, Shanggao, first described the Gougu Theorem. In the famous ancient mathematics book, 《九章算术》,the proof was given (left). The Gougu Theorem is that "in the right triangle, the sum of the squares of two right sides is equal to the square of the hypotenuse." In other words, the three sides (a, b, c) of right-angled triangle satisfies the following equation:


Where a is called 勾/Gou, b is 股/Gu, and c is 弦/Xian.
For given c, how many different positive integer solutions are there? ((a,b,c)are relatively-prime. ) It is an interesting problem.
Now, your task is to solve it.
There are several test cases; each test case contains one positive integer c (0 < c < 2^30 ) in a line. c = 0 is the end of input and need not to process.
For each test case, your program should output by following format:
Case #:
There are n solution(s).
a^2 + b^2 = c^2
…..(total n line(s) )….

Where # is the test case number starting from 1, two consecutive cases should be separated by a single blank line.
For each given c, output all solutions satisfying:
Each solution should be in one line, and in order a < b < c.
You should output n solutions by ascending order of a.

Sample Input
Sample Output
Case 1:
There are 2 solution(s).
16^2 + 63^2 = 65^2
33^2 + 56^2 = 65^2

Case 2:
There are 0 solution(s).
University of Science and Technology of China
Online Judge for ACM/ICPC
Processed in 1.8ms with 1 query(s).