All Questions
133
questions
3
votes
0
answers
90
views
On thickness of binary polynomials
OEIS A169945 introduces the concept of thickness of a polynomial as the magnitude of the largest coefficient in the expansion of the square of the polynomial. Considering the $2^{n+1}$ polynomials $p(...
- 9,592
6
votes
1
answer
300
views
Multiply an integer polynomial with another integer polynomial to get a "big" coefficient
I am new to number theory, I was wondering if the following questions have been studied before.
Given $f(x) = a_0 + a_1 x + a_2 x^2 \cdots + a_n x^n \in \mathbb Z[x]$, we say that $f(x)$ has a big ...
- 1,641
0
votes
1
answer
80
views
Parametric solution of quartic diophantine equation in three variables
How can I handle the quartic diophantine equation in three variables $x$, $y$ and $z$ $$x^4-x^2=y^2-z^2$$ in general, i.e, does exists a (three-variable) parametric solution?
What I've tried is ...
10
votes
1
answer
309
views
For any polynomial f(x) with integer coefficients, is it possible that the the greatest prime factor of f(x) can be arbitrary small?
Recently I found a problem about elementary number theory: Let $f(x)=x^2+x+1$,prove that there are infinite $n\in N$ so that the the greatest prime factor of $f(n)$ is less than $n^{1.1}$.
The answer ...
- 501
1
vote
0
answers
209
views
find the smallest integer k so that for all quadratic polynomials P with integer coefficients, one of $P(1),\cdots, P(k)$ has a 0 in base 2
Find the smallest integer k so that for all quadratic polynomials P with integer coefficients, one of $P(1),\cdots, P(k)$ has a 0 in base 2 (obviously leading zeroes don't count).
Let $\mathcal{P}$ ...
- 2,031
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
0
votes
0
answers
105
views
Perfect square equation $12\alpha^2\cdot x^3+12\alpha\cdot x^2+12\alpha\left(1-\alpha\right)\cdot x+\left(2-3\alpha\right)^2$
Well, I have the following function:
$$\text{y}\left(x\right):=12\alpha^2\cdot x^3+12\alpha\cdot x^2+12\alpha\left(1-\alpha\right)\cdot x+\left(2-3\alpha\right)^2\tag1$$
Where $\alpha\in\mathbb{N}$.
...
- 28.7k
4
votes
2
answers
575
views
Does a generalized difference of powers formula exist?
The identity
$$
\left(\frac{n+1}{2}\right)^2-\left(\frac{n-1}{2}\right)^2=n
$$
can be used to represent any number as difference of two squares. (Note that this formula gives integer values when $n$ ...
- 3,187
4
votes
0
answers
243
views
Proper divisors of $P(x)$ congruent to 1 modulo $x$
Let $P(x) $ be a polynomial of degree $n\ge 4$ with integer coefficients and constant term equal to $1$. I am interested in Polynomials $P(x) $ such that for a fixed positive integer $b$, there are ...
- 234
14
votes
1
answer
420
views
Is $x^n-\sum_{i=0}^{n-1}x^i$ irreducible in $\mathbb{Z}[x]$, for all $n$?
Let the sequence of polynomials $p_n$ from $\mathbb{Z}[x]$ be defined recursively as $$p_n(x)= xp_{n-1}(x)-1$$
with initial term $p_0(x)=1$.
Then $$p_n(x)= x^n-\sum_{i=0}^{n-1}x^i $$
Question 1: is it ...
- 3,716
3
votes
0
answers
87
views
Showing the irreducibility over $\mathbb{Z}[X]$ of polynomials similar to the cyclotomic polynomials
This question follows this other question.
Let $y$ be a natural number, $x$ a variable and $$ f(x,y):= \frac{x^{2y}-1}{x+1}$$ and
$$ g(x,y):= \frac{f(x,y)^{2y+1}-1}{(f(x,y)-1)(xf(x,y)+1)}.$$
For a ...
- 3,716
2
votes
1
answer
160
views
Irreducible polynomial divisible by all primes
Does there exist an irreducible non-linear polynomial $P(x)\in\mathbb{Z}[x]$ such that for any prime number $q$ there exists $t\in\mathbb{N}$ such that $q|P(t)$ ?
Also (dis)proving whether there ...
- 1,528
0
votes
0
answers
48
views
Polynomial with $p$ dividing $q-1$
Does there exist a non-constant polynomial $P(x)$ with integer coefficients such that for any two (positive) primes $p<q$ with $p \mid P(x_1)$ and $q \mid P(x_2)$ for some (not necessarily distinct)...
- 2,733
-2
votes
1
answer
168
views
What is the derivative of $f(n) = n^2 - 9 \bmod 16$?
I'm reading Tom M. Apostol's introduction to analytic number theory. In chapter 2 he defines
Does his definition apply to this modular polynomial in the question? I feel it does not. I'd think the ...
- 153
1
vote
0
answers
67
views
What is the maximum possiblity for an integer polynomial to be divisible by a prime number?
Description
Given a prime number $p$, a natural number $n < p$, and a random variable x from a discrete uniform distribution $\mathcal{u} = \text{unif}\{1,p-1\}$.
Let polynomial
$$Q(x) = a_{p-2}x^{...
