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) with integer coefficients that satisfy the given condition, we need to analyze the equation:
a * x * (f(x + a) - f(x)) = n * f(x)
Let's break down the steps on how to approach this problem:
1. We know that f(x) has integer coefficients, so we need to consider polynomials of the form:
f(x) = an * x^n + an-1 * x^(n-1) + ... + a1 * x + a0
2. Substitute f(x) with the polynomial form into the given equation:
a * x * ([(an * (x + a)^n + an-1 * (x + a)^(n-1) + ... + a1 * (x + a) + a0)] - [an * x^n + an-1 * x^(n-1) + ... + a1 * x + a0]) = n * [an * x^n + an-1 * x^(n-1) + ... + a1 * x + a0]
3. Simplify and group the terms accordingly:
a * x * (Sum[i=0 to n](C(n, i) * a^(n-i) * x^i) - Sum[i=0 to n](C(n, i) * a^(n-i) * x^i)) = n * Sum[i=0 to n](C(n, i) * a^(n-i) * x^i)
4. To get the maximum number of polynomials, we want the equation to hold for all x and a. Therefore, the coefficients of the terms on both sides must be equal.
5. Equate the coefficients of each power of x on both sides separately:
a * (C(n, i) * a^(n-i) - C(n, i) * a^(n-i)) = n * C(n, i) * a^(n-i)
For the above equation to be an integer, the term inside parentheses should be divisible by a.
6. Analyzing the term:
C(n, i) * a^(n-i) - C(n, i) * a^(n-i)
We notice that this term is zero, as the two terms are equal but with different signs. So, we have:
0 = n * C(n, i) * a^(n-i)
This equation tells us that either n = 0, C(n, i) = 0, or a = 0. Since we are dealing with positive integers for n, n can only be zero when the polynomial is a constant.
7. We conclude that for n to be non-zero, we need C(n, i) to be zero or a to be zero.
8. To find the maximum number of polynomials f(x), we should count the number of possible values for a and the number of possible values for i that satisfy the conditions.
9. For a fixed n, the number of possible values for a is equal to the number of positive divisors of n. This can be found by prime factorizing n.
10. For i, we have to find the maximum value that satisfies C(n, i) = 0. We can do this by checking all possible values of i and stopping when we find the largest i that satisfies C(n, i) = 0.
11. Finally, multiply the number of possible values for a by the number of possible values for i to get the maximum number of polynomials f(x) for a fixed n.
By following these steps, you can find the largest possible number of polynomials f(x) for any given n<1000.