All Questions
56
questions
1
vote
1
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39
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Can relative (or even absolute) quotient size be calculated from a list of polynomials which are multiples of a given variable?
I am working on a Diophantine equation in integers $x$ and $y$. The equation has been solved, so I already know the solutions (there are four) — I am trying to find a more elementary solution.
Through ...
- 7,929
1
vote
1
answer
86
views
Find all integer polynomials $f(x)$ such that $f(n)\mid 2^n-1$ for all $n\in\mathbb{N}^+$. [duplicate]
Find all integer polynomials $f(x)$ such that $f(n)\mid 2^n-1$ for all $n\in\mathbb{N}^+$.
So far, I have tried to plug in values of $n$, and see where that takes me. For example, plugging in $n=1$ ...
user1200609
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
3
votes
2
answers
74
views
Does $(x^2+xy+y^2)$ divide $(x+y)^n-x^n-y^n$ if and only if n has no prime factors less than 5
I noticed that for $n$ with prime factors greater than 5, that $xy(x+y)(x^2+xy+y^2)$ always seems to divide $(x+y)^n-x^n-y^n$. The $xy(x+y)$ factors seem fairly obvious, but I can't figure out where ...
- 305
6
votes
2
answers
277
views
Prove that $p(x)=x+1$.
Let $p(x)$ be a polynomial with integer coefficients such that $p(n) > n$ for every positive integer $n$. Define a sequence by $x_1 = 1, x_{i+1}=p(x_i)$ for $i\ge 1$. Suppose for any positive ...
- 1,837
1
vote
2
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275
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Find all polynomials $P(x)$ so that $P(x)(x+1)=(x-10)P(x+1)$
Find all polynomials $P(x)$ so that $P(x)(x+1)=(x-10)P(x+1)$.
I'm looking for a general solution to the above problem. For instance, say I was trying to find all polynomials $P$ satisfying $(x+1) P(x)...
- 2,031
2
votes
2
answers
199
views
Show that there exists no integer $n$ such that $n^3 - n + 3$ divides $n^3 + n^2 + n + 2$
My attempt:
For $n^3 - n + 3$ to divide $n^3 + n^2 + n + 2$, it should also divide $(n^3 + n^2 + n + 2) - (n^3 - n + 3) = n^2 + 2n - 1$. I did this to reduce the degree, but I don't think it helps.
- 404
0
votes
2
answers
107
views
Primes of the form $n^2+n+1$
Let $f(n)=n^2+n+1$. While experimenting I found that
Given $m,n\in\mathbb N, \: n>1$.
If $f(2n)\in\mathbb P$ and $f((3m+1)n)\in\mathbb P$ then $3|n$.
It has ben tested for $0\le m <1000$ and $...
- 13.9k
2
votes
2
answers
110
views
Given a polynomial $W(x)$. Prove that there exists an integer $m$ that satisfies $W(m)=W(m+1)=0$.
Given a polynomial $W(x)=x^2+ax+b$ with integer coefficients that satisfies the condition: for every prime $p$ there exists an integer $k$ that $p$ divides both $W(k)$ and $W(k+1)$. Prove that there ...
- 521
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
0
votes
1
answer
93
views
$x^d-1|x^n-1$ iff $d|n$ [duplicate]
This is an exercise that I solved before many semesters, and I just remember I solved in a very complicated(or stupid) way, today when I need to use this exercise as a tool I wonder if there are some ...
user867836
-2
votes
3
answers
440
views
polynomial of degree $n$ with real coefficient takes integral values on a certain set $n+1$ consecutive integers , then $f(x)$ is integer-valued [closed]
Show that if $f(x)$ is a polynomial of $deg n$ with real coefficient ,which takes integral values on a certain set $n+1$ consecutive integers , then $f(x)$ is integer-valued.
How to prove this I ...
- 41
2
votes
2
answers
102
views
Does $\,n\mid a_1b_1,\,a_2b_2, a_1b_2+a_2b_1\Rightarrow n\mid a_1b_2$?
Imagine that we have two pairs of integers $(a_1,b_1)$ and $(a_2, b_2)$ where
$$ a_1b_1\equiv 0,\,\ a_2b_2\equiv 0,\,\ a_1b_2+a_2b_1\equiv 0\pmod n$$
Does this imply that
$$ a_1 b_2 \equiv 0\pmod n $$
...
- 401
0
votes
5
answers
60
views
Prove or disprove this statement: For any $y \in \mathbb{Z}$ and $y \neq \pm 1$, $(y+1)^{2}$ is not divisible by $y$.
I want to prove or disprove this statement:
For any $y \in \mathbb{Z}$ and $y \neq \pm 1$, $(y+1)^{2}$ is not divisible by $y$.
The case where $y$ is even can be easily proved. However, I am stuck ...
- 3
2
votes
1
answer
79
views
Divisibility of functions
For which $n \in \mathbb{N} $ is $z^{4n}-z^{3n}+z^{2n}-z^{n}+1$ divisible by $z^{4}-z^{3}+z^{2}-z^{1}+1$
I tried factoring the second function and then fill the answer in the first function, but there ...
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