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Stack Exchange Forum » Number Theory

1h ago

Suppose we have a finite list of $n$ triples of positive integer numbers, as:
$$ \mathcal{L}=\{(a_{i1},a_{i2},a_{i3}):a_{ij}\in \mathbf{N}\setminus\{0\}, \text{ for } j=1,2,3\}_{i=1,\dots,n}.\ $$
Is there a software, where I can give $\mathcal{L}$ as in-put and which gives as out-put a possible formula, denoted by $F(a_{i2},a_{13})$, such that $a_{i1}=F(a_{i2},a_{13})$ (it could be not existing or it is not unique)? This formula can involve any operations of sum, multiplication, power, division, or also integer parts or binomial coefficients.
Just two examples to be clearer.
Consider the fol ..read more

Stack Exchange Forum » Number Theory

3h ago

On particular case of system of equation:
$\begin{cases}\sum _{cyc} x^2+y^2+xy=m^2\end{cases} $
$\begin{cases} x^2+y^2+xy=a^2\space\space\space\space\space\space\space\space (1)\\x^2+z^2+xz=b^2 \space\space\space\space\space\space\space\space (2)\\z^2+y^2+zy=c^2\space\space\space\space\space\space\space\space (3)\end{cases} $
Where $a^2-c^2=c^2-b^2$.
Solution: We have :
$a^2-c^2=c^2-b^2\Rightarrow 2c^2=a^2+b^2$
This equation has infinitely many solutions, for example $(a, b, c)=(7, 17, 13)$; we solve the system for these values:
(1)-(3): $\rightarrow (x-y)(x+y+z)=7^2-13^2=-120$
(3)-(2): $\righ ..read more

Stack Exchange Forum » Number Theory

5h ago

So the above statement is a question from a course on Theory of Computation. I have already proven it using the pumping lemma, but am looking for proofs using the Myhill-Nerode theorem, or more specifically, it's corollary, which states that a language, $L$, has a finite number of equivalence classes under the relation $\cong_L$ (indistinguishability under L), if and only if it is regular.
My idea for this was to take two strings, say $x$ and $y$, and concatenate a string, $w$ to both of these. Then, if the two strings are, indeed indistinguishable under $L$, then $|x| + |w|$ and $|y| + |w|$ s ..read more

Stack Exchange Forum » Number Theory

7h ago

I am trying to find a calculation for the total number of partitions in a euclidean space.
So, you have a 2D space. Firstly, it is partitioned into 4 parts, then, each subsequent part must be divided into 4 or 10 parts. I must find a simple way to check if the number of partitions can reach 20 000 using this method.
I am new to this, pardon me for this illustration (A denotes a node):
A
A A A A
A A A A AAAAAAAAAA A A A A A A A A ..read more

Stack Exchange Forum » Number Theory

7h ago

For any positive integer $d$ define the box $B_d$ as all points $(x_1, ..., x_d) \in \mathbb{R}^d$ with $|x_i| < 1.$
Suppose I have some invertible $n \times n$ matrix $M_1$, and for an integer $k<n$ I have the $k \times n$ projection matrix $P$ which projects a vector from $\mathbb{R}^n$ to $\mathbb{R}^k$ by removing the last $n-k$ components.
So I can look at the set $PM_1B_n$, which is transforming and then projecting the box $B_d$. Is it true that $PM_1B_n = M_2B_k$ for some invertible $k \times k$ matrix $M_2$? That is, if I transform and then project the box $B_n$, is that always e ..read more

Stack Exchange Forum » Number Theory

7h ago

Let $S$ be the set of all numbers that can be expressed as the reciprocal of the sum of two elements in $S.$ If $S$ contains $\frac{1}{2},$ characterize the exact set of numbers that are in $S.$ Note that the two elements may be identical.
In the previous post, someone showed that all elements of $S$ are in $\left[\frac{1}{2},1\right].$ I was not able to receive a good hint on how to proceed, so I am reposting and rewording the question here ..read more

Stack Exchange Forum » Number Theory

7h ago

Delete question. Delete question. Delete question ..read more

Stack Exchange Forum » Number Theory

9h ago

Find the possible remainders of $(1-3i)^{2009}$ when divided by $13+2i$ in $\mathbb{Z}[i]$.
I'm having a hard time understanding remainders in $\mathbb{Z}[i]$. I'm gonna write my solution to the problem above just to give some context, but you can skip this part and read the question below if you want. We start by noticing that $N(13+2i) = 173$, which is a prime number. Therefore, $13+2i$ is irreducible in $\mathbb{Z}[i]$. This allows us to quickly calculate the number of elements of the multiplicative group $(\mathbb{Z}[i]/(13+2i))^{\times}$, namely $N(13+2i) - 1 = 172$. By Lagrange's Theor ..read more

Stack Exchange Forum » Number Theory

14h ago

This construction comes from the chapter VII of "the local Langlands conjecture for $GL(2)$":
Let $F$ be a local field. We know that for any $n \in \mathbb{N}$ there exists a unique unramified extension of degree $n$. We can denote by $F_{\infty}$ the composite of all these fields that is clearly the maximal unramified extension of $F$. Now for each integer $n\geq 1$, $p \not\mid n$ (where $p$ is the characteristic of the residue field of $F$), the field $F_{\infty}$ has a unique extension $E_n/F_{\infty}$ of degree $n$.
My question is: why has $F_{\infty}$ a unique extension of degree $n$?
Th ..read more

Stack Exchange Forum » Number Theory

14h ago

I'm leaving this forum. Biased stance and meaningless aggressive attitudes. Good luck everyone.Please delete my questions. Take science as your truth ..read more