All Questions
12
questions
3
votes
2
answers
147
views
A conjecture about the sum of Faulhaber polynomials mod 2.
Context: In this question I asked about "Finding a polynomial function for alternating $m$ numbers of odds and even numbers in sequences" I noticed that all Faulhaber polynomials are ...
- 6,620
19
votes
2
answers
766
views
Is the smallest root of a polynomial always complex if the coefficients is the sequence of prime numbers?
The smallest root of a polynomial is defined as the root which has the smallest absolute value. Consider the polynomial $a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$. I observed that if $a_n$, is the ...
- 20.8k
1
vote
1
answer
99
views
Prove that the sequence $(a_n)_{n\ge 1}$ is unbounded
Let $f(x)$ be a non-constant polynomial with integer coefficients. For any $n\in\mathbb{N}$, let $a_n$ be the remainder when $f(3^n)$ is divided by $n$. Prove that the sequence $(a_n)_{n\ge 1}$ is ...
- 1,225
4
votes
1
answer
107
views
Show that the sequence $\{a_n\}$ is periodic.
Suppose $f(x)$ and $g(x)$ are two integer polynomials with no common complex roots. For every integer $n,$ let $a_n = \gcd(f(n),g(n))$. Show that the sequence $\{a_n\}$ is periodic.
Note that $f$ and ...
- 1,837
2
votes
0
answers
93
views
Alternating Summation of Catalan numbers $\pmod p$
Let $C_n(x)={(2n)!\over (n+1)!(n!)}$ denote the $n$-th Catalan number: $C_0=1, C_1=1, C_2=2, C_3=5, C_4=14, C_5=42, ...$
If $p$ is an odd prime, prove that
$$\sum_{n=1}^{k=(p-1)/2} (-1)^{n+1}{C_n}x^{k-...
- 3,062
-1
votes
1
answer
51
views
Show that $ 2^{ab}+1=(2^{a}+1)(2^{ab-b}-2^{ab-2b}+2^{ab-3b}+...+1) $ where b is an odd number and( a, b) are natural numbers
Show that :
$$ 2^{ab}+1=(2^{a}+1)(2^{ab-b}-2^{ab-2b}+2^{ab-3b}+...+1) $$
where b is an odd number and( a, b) are natrual numbers.
So far I’ve tried to use a similar identity as that used to prove the ...
0
votes
1
answer
82
views
show there is a number that occurs infinitely often in the digital representation of a polynomial
I would appreciate some help with the following problem:
Let $p(x)$ be a polynomial with integral coefficients (possibly negative).
Let $a_n$ be the digital sum in the decimal representation of $p(n)$...
- 807
0
votes
1
answer
33
views
Prove that sequence $S_N(a, n, d) = ±1 \pmod N$ if $N$ is prime.
The polynomial Sequence $S_N(a, n, d)$ where $N$ is the nth term in the sequence (for any integers a, n, d) is defined the recurrence relation:
$S_N = a\cdot S_{N-1}+d$
$S_{N-1} = a\cdot S_{N-2}+d$
...
- 3,062
2
votes
1
answer
97
views
If $S(n)=\sum_{i=1}^{n} u_i \in \Bbb Q$ for every $n$, do we have $S(n) = P(n)/Q(n)$ for some $P, Q\in \mathbb{Q}[x]$?
Denote $S(n)=\sum_{i=1}^{n} u_i$. I have a question that if $S(n)\in \mathbb{Q}, \forall n \in \mathbb{N}$, can I prove that $S(n)=P(n)/Q(n)$ with $P(x), Q(x)\in \mathbb{Q}[x]$, or can someone show me ...
- 743
1
vote
1
answer
65
views
Is there a proof of this process? Finding polynomial equations from a series
I'm guessing it has been but I can't find it anywhere. Would love to find a name for this process and a write up of it.
To start I know how to find the equation of a polynomial of N degree with N-1 ...
- 113
0
votes
1
answer
53
views
Can we have the sequence of polynomials that behave in this way?
Suppose that we define the sequence of polynomials in such a way that the "next" polynomial in the sequence has the same coefficients as the one "before" except that the "next" one is one degree ...
- 5,016
10
votes
1
answer
407
views
Do roots of a polynomial with coefficients from a Collatz sequence all fall in a disk of radius 1.5?
Consider a modified version of Collatz sequence:
$C(n)=\left\{ \begin{array}{ll} \frac{3n+1}{2} & n\ \mathrm{odd} \\ \frac{n}{2}& n\ \mathrm{even}\end{array} \right.$
Let $F_n$ be the ...
