All Questions
18
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Difference between polynomials with integer and non integer coefficiants regarding number of roots of polynomial [duplicate]
There is a prior question to show that polynomial of degree 7 with all integer coefficiants has 7 integer values $ P(x1,x2...x7)=+-1$ cant be expressed as product of two polynomials with integer ...
- 33
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1
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91
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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
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1
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49
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If $s = 4m - 3$, then $\sigma(p^s) = (1 + p^{2m-1})(1 + p + \ldots + p^{2m-2})$. Is there a similar factorization for $\sigma(q^t)$ when $t=4n$?
Let $p,q$ be (odd) primes, and let $m,n,s,t$ be positive integers.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
M. A. Nyblom showed that, if $s = 4m - 3$...
- 8,010
2
votes
2
answers
58
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Factorizations versus integer solutions in single-variable quartics
If $x$ is a positive integer, and I have the factorization
$$(x-1)(79x^3+159x^2-513x+255)=0,$$
what is the easiest way to conclude that $x=1$ is the only integer solution?
Related question: If [in a ...
- 7,929
2
votes
2
answers
110
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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
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3
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77
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Complete square to show that $(x-2y)^4+64xy+16 \geq 0$
I want to prove that
$f(x,y) = (x-2y)^4+64xy+16 \geq 0$
Using only algebraic methods (I guess it can be solved via completing the square).
I didn't manage to do so. Any help will be appreciated.
...
- 289
1
vote
2
answers
73
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Show that $x^4 +1 | x^{p^2-1}-1$, for prime $p>3$
I'm reading through a solution in my lecture notes that says that $x^4 +1 | x^{p^2-1}-1$, for prime $p>3$.
I've tried proving this myself but I'm struggling and not sure if it's true or if it's a ...
- 23
1
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2
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84
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Show that $7 |(n^6 + 6)$ if $7 ∤ n$, $∀ n ∈ ℤ$
Show that $7 |(n^6 + 6)$ if $7 ∤ n$, $∀ n ∈ ℤ$
I need to prove this by the little Fermat's theorem.
My attempt
$n^6 \equiv -6 \pmod 7$
To show $7 ∤ n$ I need to show that $N$ is not congruent to $...
- 637
8
votes
3
answers
2k
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Prove $n^5+n^4+1$ is not a prime
I have to prove that for any $n>1$, the number $n^5+n^4+1$ is not a prime.With induction I have been able to show that it is true for base case $n=2$, since $n>1$.However, I cannot break down ...
- 598
1
vote
4
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316
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Is $x+1$ a factor of $x^{2016}-1$?
$$x^{2016}-1=(x-1)(1+x+x^2+x^3+\dots+x^{2015})$$
If $x+1$ is a factor of $x^{2016}-1$, then $(1+x+x^2+x^3+\dots+x^{2015})=(x+1)G(x)$, where $G(x)$ is some polynomial.
What is $G(x)$ if $x+1$ is also ...
user301285
2
votes
0
answers
135
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factoring polynomials in ring of integers modulo powerful number
I am having trouble finding info on how to factor polynomials in ring of integers modulo powerful number.
For example: $x^2 - 1$ in $\textbf Z_{8}$.
I know by tinkering around that $(x - 1)(x + 1)$...
- 21
1
vote
1
answer
442
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Method to find smallest value of $x$ for which $x^2-x+C$ is composite.
Problem statement: Given the function $f(x)=x^2-x+C$, where $x$ is a positive integer $>1$ and $C$ is a positive integer ($C=0$ is also allowed), find some method and/or set of rules to always find ...
- 815
3
votes
2
answers
214
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Finding integers satisfying $m^2 - n^2 = 1111$
We have to find the integers $m$ and $n$ which will satisfy the given condition:
$$m^2-n^2=1111.$$
What could be the answer and how? i tried using trial and error and that took a long time.
- 175
5
votes
5
answers
2k
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How to factor the polynomial $x^4-x^2 + 1$
How do I factor this polynomial: $$x^4-x^2+1$$
The solution is: $$(x^2-x\sqrt{3}+1)(x^2+x\sqrt{3}+1)$$
Can you please explain what method is used there and what methods can I use generally for 4th or ...
- 1,655
0
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1
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234
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Factorization with a Primitive Factor of Polynomials
Question: Let $f,g\in\Bbb Q[x]$. Why is it that
$\rm\color{#c00}{(1)}$ if $f$ is monic then $f=\frac{1}{a}f^*$ for some primitive polynomial $f^*\in\Bbb Z[x]$ and $a\in\Bbb Z$ ?
$\rm\color{#c00}{(2)}$ ...
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