Problem p00863 - Generation 3

Orig Description

Problem J: The Teacher’s Side of Math
One of the tasks students routinely carry out in their mathematics classes is to solve a polynomial equation. It is, given a polynomial, say X2 - 4X + 1, to find its roots (2 ± √3).
If the students’ task is to find the roots of a given polynomial, the teacher’s task is then to find a polynomial that has a given root. Ms. Galsone is an enthusiastic mathematics teacher who is bored with finding solutions of quadratic equations that are as simple as a + b√c. She wanted to make higher-degree equations whose solutions are a little more complicated. As usual in problems in mathematics classes, she wants to maintain all coefficients to be integers and keep the degree of the polynomial as small as possible (provided it has the specified root). Please help her by writing a program that carries out the task of the teacher’s side.
You are given a number t of the form:
    t = m√a+n√b
where a and b are distinct prime numbers, and m and n are integers greater than 1.
In this problem, you are asked to find t's minimal polynomial on integers, which is the polynomial F(X) = adXd + ad-1Xd-1 + ... + a1X + a0 satisfying the following conditions.
 Coefficients a0, ... , ad are integers and ad > 0.
 F(t) = 0.
 The degree d is minimum among polynomials satisfying the above two conditions.
 F(X) is primitive. That is, coefficients a0, ... , ad have no common divisors greater than one.
For example, the minimal polynomial of √3+ √2 on integers is F(X) = X4 - 10X2 + 1. Verifying F(t) = 0 is as simple as the following (α = 3, β = 2).
F(t) = (α + β)4 - 10(α + β)2 + 1
= (α4 + 4α3β + 6α2β2 + 4αβ3 + β4 ) - 10(α2 + 2αβ + β2) + 1
= 9 + 12αβ + 36 + 8αβ + 4 - 10(3 + 2αβ + 2) + 1
= (9 + 36 + 4 - 50 + 1) + (12 + 8 - 20)αβ
                     = 0
Verifying that the degree of F(t) is in fact minimum is a bit more difficult. Fortunately, under
the condition given in this problem, which is that a and b are distinct prime numbers and m and n greater than one, the degree of the minimal polynomial is always mn. Moreover, it is
always monic. That is, the coefficient of its highest-order term (ad) is one.
Input
The input consists of multiple datasets, each in the following format.
   a    m   b      n
This line represents m√a + n√b. The last dataset is followed by a single line consisting of four zeros. Numbers in a single line are separated by a single space.
Every dataset satisfies the following conditions.
 m√a + n√b ≤ 4.
 mn ≤ 20.
 The coefficients of the answer a0, ... , ad are between (-231 + 1) and (231 - 1), inclusive.
Output
For each dataset, output the coefficients of its minimal polynomial on integers F(X) = adXd + ad-1Xd-1 + ... + a1X + a0, in the following format.
ad ad-1    ...   a1 a0
Non-negative integers must be printed without a sign (+ or -). Numbers in a single line must
be separated by a single space and no other characters or extra spaces may appear in the output.
Sample Input
3 2 2 2
3 2 2 3
2 2 3 4
31 4 2 3
3 2 2 7
0 0 0 0
Output for the Sample Input
1 0 -10 0 1
1 0 -9 -4 27 -36 -23
1 0 -8 0 18 0 -104 0 1
1 0 0 -8 -93 0 24 -2976 2883 -32 -3720 -23064 -29775
1 0 -21 0 189 0 -945 -4 2835 -252 -5103 -1260 5103 -756 -2183

Extracted Specification

An integer a (distinct prime number).
An integer m (greater than 1).
An integer b (distinct prime number).
An integer n (greater than 1).

### Example Input:

```
3 2 2 2
```

### Function Signature:
Write a function f(inputs) that takes in the list of tuples. 
def f(inputs: List[Tuple[int, int, int, int]]): 
    '''
    inputs: a list of tuples, each containing:
        - an integer a
        - an integer m
        - an integer b
        - an integer n
    '''

Test Case Generator

Sure! Below is a test case generator for the problem "The Teacher’s Side of Math." This generator will create valid test cases based on the constraints given in the problem statement.

```python
import random

def generate_prime(upper_limit):
    """Generate a random prime number less than or equal to upper_limit."""
    sieve = [True] * (upper_limit + 1)
    sieve[0] = sieve[1] = False
    for start in range(2, int(upper_limit**0.5) + 1):
        if sieve[start]:
            for multiple in range(start*start, upper_limit + 1, start):
                sieve[multiple] = False
    primes = [num for num, is_prime in enumerate(sieve) if is_prime]
    return random.choice(primes)

def tcgen():
    datasets = []
    while len(datasets) < random.randint(1, 5):  # Generate between 1 and 5 datasets
        a = generate_prime(100)  # Generate a prime number less than or equal to 100 for variety
        b = generate_prime(100)
        while a == b:  # Ensure a and b are distinct primes
            b = generate_prime(100)
        
        m = random.randint(2, 4)  # m should be an integer greater than 1
        n = random.randint(2, 4)  # n should be an integer greater than 1
        
        if m * n <= 20 and m * (a**0.5) + n * (b**0.5) <= 4:
            datasets.append(f"{a} {m} {b} {n}")
    
    datasets.append("0 0 0 0")  # Append the end marker
    
    return "\n".join(datasets)

# Example of generating test cases
print(tcgen())
```

Explanation:
- `generate_prime(upper_limit)`: This function generates a random prime number less than or equal to `upper_limit` using the Sieve of Eratosthenes.
- `tcgen()`: This function generates multiple datasets, ensuring that the products `m * n` and `m√a + n√b` fit within the constraints given. It generates between 1 to 5 datasets for variety.
- The function then ensures distinct primes for `a` and `b` and ensures that `m` and `n` are integers greater than 1 and their product is within the constraint (≤20). It also verifies that the sum of `m√a + n√b` is within the limit (≤4).

Each dataset is formatted as a string and collected into a list, which is eventually joined and returned as a single string with each dataset on a new line. The end marker "0 0 0 0" is appended to signify the end of datasets.

Extract Arguments

def extract_arguments(fh):
    inputs = []
    while True:
        line = fh.readline().strip()
        if line == "0 0 0 0":
            break
        a, m, b, n = map(int, line.split())
        inputs.append((a, m, b, n))
    return inputs