104 - The 11th Zhejiang University Programming Contest - G
In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The prime elements of Z[i] are also known as Gaussian primes. Gaussian integers can be uniquely factored in terms of Gaussian primes up to powers of i and rearrangements.
A Gaussian integer a + bi is a Gaussian prime if and only if either:
0 is not Gaussian prime. 1, -1, i, and -i are the units of Z[i], but not Gaussian primes. 3, 7, 11, ... are both primes and Gaussian primes. 2 is prime, but is not Gaussian prime, as 2 = i(1-i)2.
Your task is to calculate the density of Gaussian primes in the complex plane [x1, x2] × [y1, y2]. The density is defined as the number of Gaussian primes divided by the number of Gaussian integers.
There are multiple test cases. The first line of input is an integer T ≈ 100 indicating the number of test cases.
Each test case consists of a line containing 4 integers -100 ≤ x1 ≤ x2 ≤ 100, -100 ≤ y1 ≤ y2 ≤ 100.
For each test case, output the answer as an irreducible fraction.
3 0 0 0 0 0 0 0 10 0 3 0 3
0/1 2/11 7/16
Author: Local Contests 2011 Committee
Contest: The 11th Zhejiang University Programming Contest