All Questions
69
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
8
votes
1
answer
169
views
Finding a polynomial function for alternating m numbers of odds and even numbers in sequences
Context: The function $(-1)^n$ alternates between $1,-1$ because $n$ alternates between odd and even, I was trying to find a function $f(n)$ that will give $m$ positives then $m$ negatives and so on ...
- 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
106
views
$f(x)$ has a factor $(x-r)$, if and only if $f(r) = 0$ - why can't $f(r)$ be a factor of $(x-r)$?
It is clear from the polynomial remainder theorem that $(x-r)$ is a factor of $f(x)$ if $f(r) = 0$
Is there no function $f(r)$ which is a factor of $(x-r)$? does this only happen when $f(r) = 0$?
$f(x)...
2
votes
4
answers
531
views
Find the least prime $p$, such that for some positive integers $a$, $b$, $c$, $d$, $(ab + a - b)(bc + b - c)(cd + c - d)(da + d - a) = 2023^2p.$
Problem
Find the least prime $p$, such that for some positive integers $a$, $b$, $c$, $d$, $(ab + a - b)(bc + b - c)(cd + c - d)(da + d - a) = 2023^2p.$
This problem is from the algebra round of a ...
- 167
1
vote
1
answer
314
views
Find all integers $n$ such that $(n - 1)^2 + 3$ divides $n^3 + 2023$.
Problem: Find all integers $n$ such that $(n - 1)^2 + 3$ divides $n^3 + 2023$.
My Work:
$(n - 1)^2 + 3 = n^2 - 2n + 4$, which is always greater than 0 for all integers n.
Therefore, if $n^2 - 2n + 4$ ...
- 167
4
votes
2
answers
217
views
prove that $x^{20}+(1-x)^{20}-20$ is square free
Prove that $x^{20}+(1-x)^{20}-20$ is square free (i.e. has no repeated roots).
Note that the claim holds iff $p'(x)$ is coprime to $p(x)$, where $p(x)=x^{20}+(1-x)^{20}-20$. $p'(x)= 20x^{19}-20(1-x)^{...
- 2,031
0
votes
1
answer
91
views
Is the polynomial $p^{2k+2} - 4p^{k+2} + 6p^{k+1} - 4p^k + 1$ a square for integers $k \neq 1$ and $k \equiv 1 \pmod 4$?
My initial question in the present post is pretty basic:
Is the (integer) polynomial $$p^{2k+2} - 4p^{k+2} + 6p^{k+1} - 4p^k + 1$$ a square for integers $k \neq 1$ and $k \equiv 1 \pmod 4$, and ...
- 8,010
6
votes
8
answers
1k
views
If $x^{19}+x^{17}+x^{13}+x^{11}+x^{7}+x^{5}+x^{3}$ is divided by $(x^2 +1)$, then find the remainder
If the polynomial $x^{19}+x^{17}+x^{13}+x^{11}+x^{7}+x^{5}+x^{3}$
is divided by $(x^
2
+1)$, then the remainder is:
How Do I solve this question without the tedious long division?
Using remainder ...
- 445
5
votes
2
answers
217
views
Proving that $x(x^2-1)(x^2-10)=c$ cannot have five integer solutions for any real $c$
I found this question that caught my attention at MSE and I did a solution, but I suspect something is wrong with the solution.
Original problem says:
Prove that for any real values of $c$, the ...
- 14.8k
0
votes
1
answer
79
views
A problem in polynomials to show that $a_{n−1}$ and $a_n$ must be coprime [closed]
Let P(x) = $a_0x^n$ + · · · + $a_{n−1}x$ + $a_n$ be a polynomial with integer coefficients. Suppose the
equation P(x) = 0 has n distinct integer roots which are pairwisely coprime. Then prove
that $a_{...
- 11
9
votes
1
answer
250
views
Find all $n$ which $7(n^2 + n + 1)$ is perfect $4^{th}$ power.
Find all positive integer $n$ , which $7(n^2 + n + 1)$ is perfect $4^{th}$ power.
What I tried
Let $7(n^2 + n + 1) = a^4$ $\to$ $ 7 | a$ and $a$ is odd.
We then get $(n^2 + n + 1) = 343k^4$ ; $k \in \...
- 1,223
0
votes
0
answers
58
views
What is the correct notation and the correct solution?
My Question is,
Which one is a correct notation and a correct solution?
$$2x+2 \mod (x^2-x-2)=2x+2$$
Or, $$\text{RemainderPolynomial} [2x+2,x^2-x-2,x]=2x+2$$
If, both are correct why the remainder ...
user548054
0
votes
3
answers
115
views
Find sum of all natural numbers $n $ such that $(n^2+n+1)^2$ divides $1+n+n^2+...+n^{2195}$. [closed]
Update
Find sum of all $n \in \mathbb{N}$ such that $(n^2+n+1)^2$ divides $n^{2195}+n^{2194}+...+n^2+n+1$.
I have no idea. Can anyone help? At least with a hint. Thanks for the help in advance.
...
- 2,238
1
vote
3
answers
63
views
Finding number of integral solutions to an equation.
Find the number of integral solutions to:
$$x^2+y^2-6x-8y=0.$$
My attempt:
The equation can be rewritten as:
$$x^2+y^2-6x-8y+9+16=25,$$
basically adding 25 to both sides, or equivalently,
$$(x-...
- 902