Find the largest possible number of distinct integer values {x_1,x_2,…,x_n}, such that for a fixed reducible degree 4 polynomial with integer coefficients, |f(x_i)| is prime for all i?
To find the largest possible number of distinct integer values {x_1, x_2, ..., x_n}, such that |f(x_i)| is prime for all i, we first need to understand what it means for a polynomial to be reducible and how to determine if |f(x_i)| is prime for a given value of x_i.
A reducible polynomial is a polynomial that can be factored into two or more polynomials of lower degree. In the case of a degree 4 polynomial, it can be reducible if it can be factored into two polynomials of degree 2.
To determine if |f(x_i)| is prime for a given value of x_i, we need to compute the value of the polynomial at that particular value and then check if the absolute value of the result is a prime number. If it is prime, then |f(x_i)| is prime; otherwise, it is not.
Now, to find the largest possible number of distinct integer values {x_1, x_2, ..., x_n} meeting our requirements, we can follow these steps:
1. Start by listing all possible integer values for x_i, starting from a minimum value and ending at a maximum value. The specific range of values depends on the problem or constraints given.
2. For each integer value of x_i, compute the value of the polynomial f(x_i).
3. Check if the absolute value of f(x_i) is prime using a primality test. One common primality test is checking if f(x_i) is divisible by any prime number less than or equal to the square root of f(x_i). If it is not divisible, then f(x_i) is prime.
4. Keep track of the number of distinct integer values {x_1, x_2, ..., x_n} for which |f(x_i)| is prime.
5. Repeat steps 2-4 for all integer values of x_i.
6. Once you have checked all possible integer values of x_i, you will have the largest possible number of distinct integer values {x_1, x_2, ..., x_n} that satisfy the given conditions.
It is important to note that determining the number of distinct integer values may require checking a large number of values, depending on the specific polynomial and chosen range for x_i. Additionally, it is recommended to use efficient primality tests to speed up the process when checking if |f(x_i)| is prime.