Call a polynomial $f(x)$ excellent if its coefficients are all in [0, 1) and $f(x)$ is an integer for all integers $x$. a) Compute the number of excellent polynomials with degree at most 3. b) Compute the number of excellent polynomials with degree at most $n$, in terms of $n$. c) Find the minimum $n\ge3$ for which there exists an excellent polynomial of the form $\frac{1}{n!}x^n+g(x)$, where $g(x)$ is a polynomial of degree at most $n-3$.
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Tags: Comc, algebra, polynomial
16.12.2024 16:33
Did anyone manage to get an idea or hint to this problem? If yes, what was the initial hint? Could you share your idea with me?
17.12.2024 06:05
An attempt... Please let me know what to fix: Let $f(x)=a_3x^3+a_2x^2+a_1x+a_0$. $f(0)=a_0$ so $a_0 \in {\mathbb{Z}}$. Similarly, for integers $x$, $a_3x^3+a_2x^2+a_1x \in {\mathbb{Z}}$. Clearly, $a_1, a_2, a_3$ are rational. Let $a_s=\frac{p_s}{q_s}$ in simpliest form. Note that if $x\nmid{q_s}$ for all $s$, then the expression will not be an integer. Therefore $q_1=q_2=q_3$. Additionally, this implies $x\nmid{q}$, $p_3x^2+p_2x+p_1 \equiv 0\pmod{q}$. Now, we need $q\leq{3}$ since there are 3 variables of $p$, and to solve the system with all residues $x$ mod $q$, we can have at most $4$ scenarios. Before proceeding, I want to check what I have so far?
19.12.2024 08:08
I like your idea to part A, I would highly use the hint below Try letting f(x)=ax^3+bx^2+cx Step1. find f(1) and f(-1), then add f(1) and f(-1) together to find all possible value of b. Step 2. Find f(1) minus f(-1) to find all possible value of a+c. I hope this helps.
19.12.2024 10:52
Mathloveguy12 wrote: Did anyone manage to get an idea or hint to this problem? If yes, what was the initial hint? Could you share your idea with me?