For every positive integer n, consider all monic polynomials f(x) with integer coefficients, such that for some real number a

x(f(x+a)−f(x))=nf(x)
Find the largest possible number of such polynomials f(x) for a fixed n<1000.

To find the largest number of monic polynomials, let's start by understanding the given equation and how to manipulate it.

Given the equation:
x * (f(x + a) - f(x)) = n * f(x)

First, let's simplify this equation by expanding f(x + a):

x * (f(x) + f'(x) * a - f(x)) = n * f(x),
x * f'(x) * a = n * f(x),
f'(x) * a = (n * f(x)) / x. (1)

Now, let's analyze the problem step by step:

Step 1: Identify key factors and limitations.
- We need to find monic polynomials with integer coefficients, which means the leading coefficient should be 1.
- We are only considering positive integers n, where n < 1000.

Step 2: Simplify equation (1).
Since we are looking for polynomials with integer coefficients, we know that f(x) and f'(x) must have integer coefficients as well. Therefore, the right-hand side, (n * f(x)) / x, should be an integer. This gives us an important condition:

n is a multiple of x. (2)

Step 3: Find all possible values of x and n.
Since we have the limitation that n < 1000, we need to find all possible x values such that n is a multiple of x and n < 1000.

Let's calculate the divisors of all numbers from 1 to 1000. For example, for n = 1000, the divisors are: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, and 1000.

Step 4: Construct the monic polynomials.
For each found divisor x, construct the monic polynomial f(x) by substituting x into equation (1) and solving for f.

For example, let's consider x = 2:
f'(2) * a = (n * f(2)) / 2.

We know that a needs to be an integer since f'(x) and f(x) have integer coefficients, and n is divisible by x. Therefore, we can find all possible values for a.

Repeat this process for each divisor x you found in step 3.

Step 5: Count the number of polynomials.
Count the number of unique monic polynomials you obtained in step 4. This will give you the largest possible number of polynomials for a fixed n.

By following these steps, you can find the largest number of monic polynomials f(x) for a fixed n < 1000.