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5
questions
1
vote
1
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170
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Method to find if a polynomial has irrational roots?
Question
Let's say I define a polynomial $P(x)$ whose roots $\alpha_i$ and degree $> 1$.
We also add constraint that the coefficient of the highest power of $P(x)$ is $1$ and all other coefficients ...
- 1,177
3
votes
2
answers
286
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When is $y=3x^2+3x+1$ a prime number in $\mathbb{Z}$ with $x \in \mathbb{Z}$?
The first few values of $y=3x^2+3x+1$ for integer values of $x$ are $7, 19, 37, 61, 91$, and $127$. I am wondering under what conditions of $x$ is $y$ a prime number?
I had initially hoped that ...
3
votes
3
answers
815
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Prove that the factorization of $x^n-1$ contains only 2 factors if and only if $n$ is prime
I would like to prove the following statement.
If $n$ is a prime number, then the factorization of $x^n-1$ over
$\Bbb{Z}$ contains only $2$ factors, and those factors are:
$$x^n-1=(x-1)(x^{n-...
- 1,598
7
votes
1
answer
403
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GCD of $n^a\,\prod\limits_{i=1}^k\,\left(n^{b_i}-n\right)$ for $n\in\mathbb{Z}$
Let $a$ be a nonnegative integer. For a given positive integer $k$, let $b_1,b_2,\ldots,b_k$ be odd integers greater than $1$. Using this result, it can be shown that, for each integer $n$, $$f_{a;...
- 49.8k
2
votes
0
answers
76
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Polynomials producing Carmichael numbers
It is well known that $$(6n+1)(12n+1)(18n+1)$$ is a Carmichael number, if all factors are prime. I found the polynomials $$(6n+1)(12n+1)(18n+1)(36n+1)$$
$$(18n+1)(36n+1)(108n+1)(162n+1)$$ $$(20n+1)(...
- 85.1k