All Questions
13
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
43
views
Maximal divisor of pairwise differences
Let $r \in \mathbb{N}$ be fixed. Define the function:
$$f : \mathbb{Z}^r \to \mathbb{N}, \quad (a_1, ..., a_r) \mapsto \max_{n \in \mathbb{Z}} \prod_{i<j} \gcd((n-a_i), (n-a_j))$$
It is perhaps not ...
- 2,466
5
votes
1
answer
126
views
show that n is a power of 2 given it satisfies a combinatorial property
Let $\{a_1,\cdots, a_n\}$ and $\{b_1,\cdots, b_n\}$ be two distinct sets of positive integers such that any integer can be written as $a_i+a_j$ with $i\neq j$ in exactly as many ways as it can be ...
- 1,837
1
vote
1
answer
73
views
Prove or disprove that $\sum_{k=1}^p G(\lambda^k) = ps(p)$
Prove or disprove the following: if $\lambda$ is a pth root of unity not equal to one, $G(x) = (1+x)(1+x^2)\cdots (1+x^p),$ and $s(p)$ is the sum of the coefficients of $x^n$ for $n$ divisible by $p$ ...
- 1,225
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
0
votes
2
answers
75
views
Solution mod 2 of polynomial equation
Can one find a polynomial $p(x,y)$ such that it is integer for integer $x,y$ and it satisfies
$$ p(x,y) + p(y,x) = x^2 + y^2 +1 \mod 2$$
or prove that it is not possible?
- 2,390
2
votes
1
answer
96
views
Coefficients of the expansion of $\prod_{i=1}^k(x+i)$
This seems to be something well known, but I couldn't find any reference.
Suppose that we wish to expand the product $\prod_{i=1}^k(x+i)$ as $a_0x^k+a_{1}x^{k-1}+\ldots+a_{k-1}x+ a_k$. The ...
- 1,337
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
3
votes
1
answer
134
views
Variable transformation $x\to x+1$ in Fibonacci-polynomial $F_n(x)$
I am seeking for an expression for the transformation of the Fibonacci-polynomials $F_n(x)$ from $x\to x+1,\; x\in\mathbb{R}$, ideally in terms of $F_m(x), m\le n$.
I have tried $$ F_n(x+1):= \sum_{k=...
- 1,715
2
votes
0
answers
69
views
Is there accepted notation for the set of ways of partitioning a natural number $a$ into $b$ parts?
Recall the following:
Multinomial Theorem. For all finite sets $X$, we have:
$$\left(\sum_{x \in X} x\right)^n = \sum_{a}[a](X)^a$$
where $a$ ranges over the set of partitions of $n$ into ...
- 68.1k
0
votes
1
answer
109
views
Number Theory and p-Remainder Numbers
In order to submit the problem, here it comes the definition we are interested in. Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$ and some natural $p > 1$, we will designate a p-remainder ...
- 21.7k
4
votes
2
answers
650
views
How to prove ${{pm} \choose {pn}}\equiv{m \choose n} \pmod{p}$.
Question:(1) if p is a prime and m,n $\in$ N,prove that ${{pm} \choose {pn}}\equiv{m \choose n} \pmod p$ (the book gives me a hint: think about $(1+x)^{pm}$ and $(1+x^m)^p$ in $F_{p}(x)$.
(2) Prove ...
- 3,500