For every positive integer n, consider all 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?

Question

For every positive integer n, consider all 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?

Answer 1

To find the largest possible number of polynomials f(x) for a fixed positive integer n < 1000, we can use the concept of divisibility.

Given the equation x * (f(x + a) - f(x)) = n * f(x), we can rewrite it as f(x + a) - f(x) = (n / x) * f(x) by dividing both sides by x.

Now, let's consider the polynomial f(x) = c, where c is a constant integer. Substituting this value in the equation, we get c - c = (n / x) * c, which simplifies to 0 = n, indicating that n must be 0 for this polynomial to satisfy the condition.

Since n is a positive integer and we are looking for polynomials f(x) with integer coefficients, we can conclude that the constant polynomial f(x) = c is not a valid solution for any n > 0.

Thus, we need to consider non-constant polynomials.

Let's assume f(x) has degree m. In order to satisfy the given equation, f(x + a) - f(x) must have a degree of m as well. Therefore, f(x) must have at least degree m + 1 for the equation to hold.

Using the fact that f(x) has integer coefficients, we can determine the number of possible polynomials f(x) for a fixed n by considering the divisibility of n by different powers of prime numbers.

For example, if n is divisible only by a single prime number p (where p^(k-1) < n < p^k), then we can have up to k polynomials of degree k for a fixed n.

To find the answer to the original question (the largest possible number of polynomials f(x) for a fixed n < 1000), we need to examine the prime factorization of n. By checking the divisibility of n by different powers of prime numbers, we can determine the maximum number of possible polynomials for each prime factor. Finally, we multiply these numbers together to get the overall maximum number of possible polynomials for a fixed n.

This process may be time-consuming for large values of n, but it provides a systematic way to find the answer by analyzing the divisibility properties of n.

For every positive integer n, consider all polynomials f(x) with integer coefficients, such that for some real number a x*(f(x+a)−f(x))=n*f(x) Find the largest possible number of such polynomials f(x) for a fixed n<1000?

To find the largest possible number of polynomials f(x) for a fixed positive integer n < 1000, we can use the concept of divisibility.

For every positive integer n, consider all 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?