All Questions
17
questions
-6
votes
0
answers
33
views
How is this rough draft of a proof that $(x^2+10x-3=0)∉Qred? [closed]
I wanted to prove that the above quadratic is irreducible over Q without using common methods such as quadratic formula, discriminate, completing the square, rational root theorem, Eisensteins ...
0
votes
0
answers
49
views
Can you check this proof? Find all polynomials $P(x)$ with integer coefficients such that for each natural number $n, n$ divides $P(2^n)$ [duplicate]
Is this a good proof? I think it is allright but I am not sure. I want to prove this polynomial must be constant, with P(x)=0.
Suppose P isn't constant Let ord_p(2) denote the smallest number 0<k ...
- 27
9
votes
2
answers
831
views
If $P(0)$ and $P(1)$ are both odd, show that $P(x)$ has no integer roots
Here is a question from Canada MO:
Let $P(x)$ be a polynomial with integer coefficients. If $P(0)$ and $P(1)$ are both odd, show that $P(x)$ has no integer roots.
My idea to solve the problem is ...
- 3,966
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
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
38
views
Given that $n=\prod^r_{i=1}p_i^{k_i}$, find $|\{x\in \mathbb{Z}/n\mathbb{Z}: x^a\}|$.
Given that $$n=\prod^r_{i=1}p_i^{k_i}$$
where $k_i\in \mathbb Z_+$ and $p_i$ is prime $\forall i\in\mathbb Z_+<r$, find $$|\{x\in \mathbb{Z}/n\mathbb{Z}: x^a\}|$$
(Note: This is a proof ...
- 3,204
1
vote
1
answer
45
views
Solution verification: family of polynomials, $\gcd$ of polynomials
Let $P$ be a family of polynomials $\mathcal P:=\{p_n(x):n\in\mathbb
N\}$ $$p_n(x)=x^3+2(1-n)x^2+n(n-4)x+2n^2$$ Let $f(x)\in\mathbb R[x]$
s.t. $\;\gcd(p_n(x),f(x))=d(x),\;\forall n\in\mathbb N.$
...
- 4,670
3
votes
1
answer
104
views
Solution verification: Prove $f$ doesn't take the value $14$ for any integer input. [duplicate]
Let $f\in\mathbb Z[x]$ be a polynomial with integer coefficients, s.t.
it takes the value $7$ for $4\;\text{distinct integers}$. Prove $f$
doesn't take the value $14$ for any integer input.
My ...
- 4,670
1
vote
3
answers
422
views
Prove that there exists a polynomial p(x) with coefficients belonging to the set {-1, 0, 1} such that p(3) = n, for some positive integer n
Prove that there exists a polynomial p(x) with coefficients belonging to the set {-1, 0, 1} such that p(3) = n, for some positive integer n.
I started off my proof by noticing that n = either 3k or ...
- 93
1
vote
0
answers
30
views
Solutions to cubic and higher degree functions
If $u = ax^2+a_2x+a_3$ is a quadratic polynomial then there exists a solution to $w=v^2$ where $w=cu+c_2$ and $v = bx+b_2$, which can be easily shown to be true.
For instance when $u=x^2+x+1$, $w=4u-...
- 3,062
3
votes
2
answers
87
views
Find the last three digits of $p$ if the equations $x^6 + px^3 + q = 0$ and $x^2 + 5x - 10^{2013} = 0$ have common roots.
Find the last three digits of $p$ if the equations $x^6 + px^3 + q = 0$ and
$x^2 + 5x - 10^{2013} = 0$ have common roots.
Let $a,b $ be the solutions of second equation, then by Vieta we have $a+...
- 90.7k
8
votes
0
answers
96
views
Prove that there do not exist distinct integers $a,b,c$ and polynomial $P$, with integer coefficients, such that $P(a)=b, P(b)=c, P(c)=a$. [duplicate]
Let $P(x)$ be a polynomial of degree $n$ with integer coefficients. Assume that there exist three distinct integers $a,b,c$ such that $$P(a)=b, P(b)=c, P(c)=a.$$
Since the integers $a,b,c$ are all ...
- 1,348
0
votes
1
answer
105
views
Proof Verification: Infinitely many Primes Using Euclid's Algorithm
I'm trying to prove the infinitude or primes using division algorithm. Does the following proof work:
Assume that there are only finitely many primes in $\mathbb{Z}$. By letting $\mathcal{P}$ denote ...
- 1,592
5
votes
1
answer
2k
views
Number Theory: Prove that $x^{p-2}+\dots+x^2+x+1\equiv 0\pmod{p}$ has exactly $p-2$ solutions
I just completed this homework problem, but I was wondering if my proof was correct:
If $p$ is an odd prime, then prove that the congruence $x^{p-2}+\dots+x^2+x+1\equiv 0\pmod{p}$ has exactly $p-2$ ...
- 1,165
6
votes
2
answers
136
views
$a,b,c\in \Bbb Z$ and $a\cdot b\cdot c$ is a root of $ax^2+bx+c$.
I was curious if there are quadratic equations where $a,b,c\in \Bbb Z$ and $a\cdot b\cdot c$ is a root of $ax^2+bx+c$.
So trivially if $c=0$, $a$ and $b$ can be arbitrary, and if either $a$ or $b$ is ...
- 4,395