All Questions
20
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
2
votes
0
answers
69
views
Guessing algebraic number by its binary digits
Positive integers $d$, $H$ are given. It's known that $r\in(0,1)$ is such that $p(r)=0$ for some nonzero $p\in\mathbb Z[x]$ with $\deg p\leq d$ and coefficients $c_i$, such that $|c_i|\le H$, $i=0,\...
- 235
3
votes
1
answer
121
views
Is there an injective polynomial $f: \mathbb{N}^{2}/S_2 \longrightarrow \mathbb{N}$?
Is there a polynomial $f: \mathbb{N}^{2} \longrightarrow \mathbb{N}$ injective except for the $𝑆_2$ action?
This polynomial must be invariant under $S_2$ action, i.e $f(x,y)=f(y,x)$. However, if $(x,...
- 160
0
votes
2
answers
113
views
How many completely reducible polynomials of degree $n$ are there in $\mathbb{F}_p$?
What do we know about the count of completely reducible polynomials modulo $p$? In other words, polynomials that factor into all linear factors and nothing of higher degree.
From this site and ...
4
votes
0
answers
362
views
Zeros of Faulhaber's formula
I've noticed that $f_p(n) := \sum_{k=1}^nk^p$ has something interesting going with its zeros:
$$
\begin{align}
p = 0: &&0
\\
p = 1: &&-1, 0
\\
p = 2: && -1, -\frac12,0
\\
p = 3:...
- 36.2k
1
vote
0
answers
52
views
Lower bound on the number of distinct polynomials of degree less than $t$
In the Primes in P paper there is a section which states that there are at least $t+l \choose t-1$ distinct polynomials of degree $<t$ in a certain group. I think this can be deduced only from the ...
- 41
2
votes
1
answer
92
views
closed formula for the function
Suppose function $f$ is such that for any $x\in R_+$ we have that
$$
f(\, x(x-1)\ldots(x-k+1)\, )=A_k.
$$
Moreover, the following holds for all $k$:
$$
f(x)=A_1\\
f(x(x-1))=f(x^2-x)=f(x^2)-f(x)=A_2\\
...
- 311
7
votes
3
answers
272
views
Some Combinatorics and Some Prime Numbers
Problem statement: Let $U=\{1,2,...n\}$ and $S$ be the set of all permutations of the elements of $U$. For any $f \in S$ let $I(f)$ denotes the number of inversions (see remark) of $f$. Let $A_j$ ...
- 6,134
3
votes
1
answer
104
views
Interesting equation $P(x)=Q(y)$, with infinite integer solutions $(x,y)$
An equation $P(x)=Q(y)$ is called Interesting if $P$ and $Q$ are polynomials with degree at least one and integer coefficients and the equation has an infinite number of answers in $\mathbb{N} \times \...
- 1,705
2
votes
1
answer
133
views
Unique Structure in base for powers $1,2$ and $3$
Let's $1<a\in\mathbb{N}$
And $$A^{k}=\sum_{i=1}^{a}i^{k}$$
Here $t $ is a number from any base $q$ can be converted in base $b$ written as
$$(t)_{q}=(b_{r} b_{r-1} ... b_{2} b_{1})_{b}$$
Now ...
- 2,697
4
votes
1
answer
383
views
Integral identity involving Bernoulli polynomials
I found the following identity on Wikipedia, and I am having a difficult time proving it.
For $m,n\in\Bbb N$,
$$I(m,n):=\int_0^1B_n(x)B_m(x)\mathrm{d}x=(-1)^{n-1}\frac{m!n!}{(m+n)!}b_{n+m}$$
Where $...
- 17.3k
1
vote
0
answers
49
views
Is there accepted notation for the various "duals" of a polynomial? Does any interesting theory surround them?
Let $k$ denote a base field. There's a variety of interesting ways to turn polynomials in $k[x]$ into a "dual operator" that acts maps $k$-linearly from $k[[x]]$ to another $k$-module, such as $k[[x]]$...
- 68.1k
1
vote
1
answer
59
views
$N_k$ is the number of pairs $(a,b)$ of non-negative integers such that $ka+(k+1)b=n+1-k$. Find $N_1+N_2+\cdots N_{n+1}$.
Let $n$ be a positive integer. Assume that:
$N_k$ is the number of pairs $(a,b)$ of non-negative integers such that $ka+(k+1)b=n+1-k$.
Find $N_1+N_2+\cdots N_{n+1}$.
I was trying to solve this ...
user545623
2
votes
1
answer
93
views
A bound on the number of zeroes of a polynomial in a given set.
Let $A_1$, $A_2$,...., $A_n$ be subsets of a field $F$ such that $|A_i| = N$ for all $i$. If $P$ is nonzero $n$-variable polynomial over $F$ of total degree $D$, show that the number of zeroes of $P$ ...
- 710