Questions tagged [cubics]
This tag is for questions relating to cubic equations, these are polynomials with $~3^{rd}~$ power terms as the highest order terms.
222
questions with no upvoted or accepted answers
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votes
0
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Generalizing Ramanujan's cube roots of cubic roots identities
(This extends this post.) Define the function,
$$\sqrt[3]{G(t)} = \sqrt[3]{t+x_1}+\sqrt[3]{t+x_2}+\sqrt[3]{t+x_3}\tag1$$
where the $x_i$ are roots of the cubic,
$$x^3+ax^2+bx+c=0\tag2$$
While $G(t)...
- 54.9k
12
votes
0
answers
910
views
The probability that a random (real) cubic has three real roots
We can formalize the notion of the probability that a randomly selected quadratic real polynomial has real roots as follows:
Suppose $R > 0$, and suppose the random variables $a, b, c$ are (...
- 102k
10
votes
0
answers
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On the solvable octic $x^8-44x-33 = 0$ and the tribonacci constant
I had discussed the solvable octic trinomial,
$$x^8-44x-33=0\tag1$$
way back in this old MSE post, but I revisited this inspired by another solvable octic,
$$y^8-y^7+29y^2+29=0\tag2$$
which I also ...
- 54.9k
9
votes
0
answers
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Omar Khayyam and the tribonacci constant
While trying to find the tribonacci cousin of this post, I came across this nice short article A Geometric Problem of Omar Khayyam and its Cubic by Wolfdieter Lang. Given the figure,
$\hskip1.7in$
...
- 54.9k
7
votes
0
answers
149
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Cubic surfaces and 27 lines
It is well known that a smooth cubic surface will contain 27 lines. All the references I came across gives a proof of the fact for fields of Characteristic $\neq 2$. Is there a reference where the ...
- 43
6
votes
1
answer
227
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The asymptotic of the number of integers that are sums of three nonnegative cubes
Let $c(n) $ be the number of distinct integers between $0 $ and $n $ of the form $ a^3 + b^3 + c^3$, meaning the sum of $3$ nonnegative cubes.
$C(n) = O( n \space \ln(n)^x ) $
Find and prove the ...
- 16.4k
5
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0
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There is a compass-like tool that can draw $y=x^2$ on paper. Is there one for $y=x^3$?
Is there a tool that can draw $y=x^3$ on paper?
I'm referring to low-tech tools, e.g. not computers.
I only know of tools that can draw $y=x^2$. The YouTube video "Conic Sections Compass" ...
- 25.7k
5
votes
0
answers
83
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Involution on $2\times 2$ matrices
Show that the map on $2\times 2$ matrices
\begin{eqnarray}
\left( \begin{matrix} a & b\\ c & d \end{matrix} \right)\overset{\Phi}{\mapsto} \left( \begin{matrix} a & b\\ c & d \end{...
- 55k
5
votes
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Factorization of $x^3-x-1$ over $\mathbb{F}_p$
Factoring quadratic polynomials over finite fields can easily be done by determining if the discriminant is a quadratic residue modulo characteristic in question, and if so, apply the quadratic ...
- 405
5
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0
answers
109
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Counterexample to the Hasse principle of the form $x^3 + y^3 + z^3 + nt^3$
Selmer's cubic is a counterexample to the Hasse principle for ternary cubic forms.
We also know that the Hasse principle does not hold for quaternary cubic forms, as
$$
5x^3 + 12y^3 + 9z^3 + 10t^3
$$
...
- 2,907
5
votes
0
answers
271
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$\sqrt[3]{\text{something}\pm\sqrt{\text{something}}}$
$$\sqrt{A\pm\sqrt{B}}=\sqrt{\frac{A+C}{2}}\pm\sqrt{\frac{A-C}{2}}$$
, where
$C^2=A^2-B$.
But, I couldn't find a formula for
$\sqrt[3]{A\pm\sqrt{B}}$.
First of all, why would you even need this? I've ...
- 741
5
votes
0
answers
714
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Elementary approach to solving cubic equations over finite fields
I am interested in studying cubic equations over finite fields. For example, when does
$$
ax^3 + bx^2 + cx + d = 0
$$
have a solution in $\mathbb{F}_q$ for $a,b,c,d\in \mathbb{F}_q$ (finite field of ...
- 3,309
5
votes
0
answers
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Which primes $\mathfrak{p}$ can be written as sums of two squares in $\mathbb{Q}(\sqrt[3]{2})$?
which primes $\mathfrak{p}$ can be written as sums of two squares in $\mathbb{Q}(\sqrt[3]{2})$? I am asking to solve an equation:
$$ \mathfrak{p} = \big(a_1+a_2\sqrt[3]{2}+a_3\sqrt[3]{4}\big)^2
+ \...
- 24.5k
5
votes
1
answer
981
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Integer solutions of a cubic equation
With $\mathrm {gcd}(x,y)=1$ I have the following equation: $$x^3-xy^2+1=N$$
I want to find the integer solutions, given an N, of the variables $x$ and $y$. I have tried factoring the equation into $x(...
- 7,348
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votes
0
answers
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A cute way to solve the quadratic. How to extend it to the cubic?
Playing around with complex numbers, I found a cute way of solving the quadratic equation.
Let's start with the (monic) equation
\begin{equation}
z^2+pz+q = 0
\end{equation}
where $z$ and the ...
- 3,386