All Questions
Tagged with polynomials elementary-number-theory
668
questions
2
votes
1
answer
388
views
Find and prove an upper bound on the number of intersections on two distinct polynomials
Find and prove an upper bound on the number of times that two distinct polynomials of degree $d$ can intersect.
What if the polynomials' degrees differ?
My attempt:
let $p(x)$ and $q(x)$ be two ...
- 2,060
11
votes
2
answers
5k
views
Proving a polynomial $f(x)$ composite for infinitely many $x$
Let $f(x)=a_0+a_1x+ \ldots +a_nx^n$ be a polynomial with integer
coefficients, where $a_n>0$ and $n \ge 1$. Prove that $f(x)$ is
composite for infinitely many integers $x$.
I can easily show ...
- 3,299
1
vote
1
answer
202
views
How to find such $a \in \mathbb{N} \setminus \{4\}$ that $x^2+ax+a$ is composite for all $x$?
In my previous questions it is shown that $f(x)=x^2+ax+a$ , where $a\in\mathbf{Z^+}$\ $\left \{ 4 \right \}$ is irreducible and that gcd$(f(1),f(2),f(3).....)=1$
So, according to Bunyakovsky ...
- 12.9k
0
votes
2
answers
89
views
$\gcd(P(a),Q(a),R(a),S(a),T(a))=1$ for any particular value of $a$?
Let's define five binomials as :
$P(a)=2a+1$
$Q(a)=3a+4$
$R(a)=4a+9$
$S(a)=5a+16$
$T(a)=6a+25$
How to prove that :
$\gcd(P(a),Q(a),R(a),S(a),T(a))=1$
for any particular value of $a$ , $(a\in \mathbb{...
- 12.9k
3
votes
3
answers
801
views
Divisors of all values of polynomial over $\Bbb Z\,$ (fixed divisors)
From Fundamentals of Number Theory by LeVeque, section 3.1, prob. 1
Let $f(x) = a_0x^n + \cdots + a_n$ be a polynomial over Z. Show that if $r$ consecutive values of $f$ (i.e., values for consecutive ...
- 593
4
votes
4
answers
840
views
Show that $q(n)=11n^2 + 32n$ is a prime number for two integer values of $n$
Let $n$ be an integer and show that $q(n)=11n^2 + 32n$ is a prime number for two integer values of $n$, and is composite for all other integer values of $n$.
- 875
4
votes
1
answer
237
views
What is the formula for this function $f(x) = (x-1)(x-2)(x-3) \cdots (x-k)$
I wonder if there exists a formula for this function?
$$f(x) = (x-1)(x-2)(x-3) \cdots (x-k)$$
I want to know the coefficient of each $x^i$, and the first thing I came up with was to find the expansion ...
- 12.2k
4
votes
2
answers
277
views
The equation $F(x) \equiv 0 \pmod m$ has integer solution for x
Let $F(x)=(x^2-17)(x^2-19)(x^2-323)$ and let $m$ be a positive integer. How can one show that the equation $F(x) \equiv 0 \pmod m$ has an integer solution?
- 4,896