All Questions
11
questions
1
vote
0
answers
550
views
Can we find an integer function satisfying these conditions?
I have the following problem:
Determine a function $f(x,y)$ defined on the positive integers and satisfying each of the following:
(1) $f(x, y)$ is a positive integer
(2) $f(2,3)=31$ and $f(4,1)=17.$
(...
- 13.4k
3
votes
1
answer
65
views
Find the term that will have the larger coefficient
Which of the expressions $$(1+x^2-x^3)^{100} \textrm{or}\:\: (1-x^2+x^3)^{100}$$ has the larger coefficient of $x^{20}$ after expending abd and collecting terms.
I can easily do this question via ...
- 1,253
2
votes
1
answer
172
views
Prove that there is a number that is not square-free
How do I prove that for any polynomial $f(x) \in \mathbb{Z}[x], \operatorname{deg} f \geq 1$, there are infinitely many $n$, such that $f(n)$ is not squarefree?
I have two solutions, one of them is ...
- 2,344
0
votes
1
answer
569
views
Proof that a-b divides f(a) - f(b). [duplicate]
Given a polynomial function f(x), prove that f(a) - f(b) is divisible by a - b. I tried it with a few examples and it is coming out to be true. Is it true and if so how do we prove it ? Thanks
- 1
8
votes
2
answers
133
views
Polynomial that indicates whether or not $x = 1 \pmod n$.
For each $n$, is there a polynomial that takes representatives $x \in \Bbb{Z}$ for $\bar{x} \in \Bbb{Z}_n$, and returns whether or not $x = 1 \pmod n$?
For example, $n = 2$. Then $x \mapsto x \pmod{...
6
votes
3
answers
106
views
What's the degree of $\binom{x}{k}$?
Consider $\binom{x}{k}$, where $x$ is a positive integer variable and $0 \leq k \leq x$, where $k$ can be dependent on $x$. I'm interested in expanding $\binom{x}{k}$ in $x$.
For 'small' values of $k$...
- 5,679
4
votes
4
answers
251
views
Find all polynomials $\mathcal{f}$ such that $\mathcal{f}$ has coefficients $\in \mathbb{N}_0$ and $\mathcal{f}(1)=7, \mathcal{f}(2)=2017$.
From (2018) Hong Kong TST 1 P2 Functions:
Find all polynomials $f$ such that $f$ has coefficients $\in \mathbb{N}_0$ and $f(1)=7, f(2)=2017$.
May I get some hints, instead of a complete solution?
- 3,334
5
votes
4
answers
1k
views
Find all functions of the form $f(x)=\frac{b}{cx+1}$ where $f(f(f(x)))=x .$
This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with functions and polynomials, but other than that, the textbook gave no hints ...
- 2,433
0
votes
1
answer
35
views
Factors/divisibility of monotonically-increasing integer polynomial
For positive integers $n$ and $x$, let $f_n(x)$ be a polynomial in $x$ of degree $n-1$, such that $f_n(x)$ is monotonically increasing for increasing $x \ge 1$.
Now assume that there exist positive ...
- 7,929
1
vote
1
answer
890
views
A polynomial is called a Fermat's polynomial...
A polynomial is called a Fermat polynomial if it can be written as the sum of the squares
of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such
that $f(0) = ...
- 6,580
8
votes
6
answers
312
views
Representing the function $\mathbb Z_9\to\mathbb Z_9$, $f(0) = 1$, $f(1) = \ldots = f(8) = 0$ as a polynomial in $\mathbb Z_9[x]$
Let $\mathbb Z_9=\left\{0,1,2,3,4,5,6,7,8\right\}$ be the set of integers modulo 9 and $f:\mathbb Z_9 \rightarrow \mathbb Z_9$ be a function.
Assume $f(0)=1$, $f(1)=f(2)=...=f(8)=0$. What is the ...
- 335