Primitive Definition and 145 Threads

Questions regarding Primitive Unit Cell (and what I think the answer are, correct me if I am wrong) 1. Can there be more than one Primitive Unit cells for the same crystal? yes, Wigner Seitz cell always will exist. There can be other primitive Unit cells along with Wigner Seitz too. But...

Hey! :giggle: How can we calculate the number of natural numbers between $2$ and $n$ that have primitive roots? Let $m$ be a positive integer. Then $g$ is a primitive root modulo $m$, with $(g,m)=1$, if the modulo of $g\in (Z/m)^{\star}$ is a generator of the group. We have that $g$ is a...

Hi guys , I want to construct a primitive unit cell for diamond, which is made of a fcc lattice and a basis of 2 carbons atoms. I know that a primitive unit cell isn't unique but the two variants I get are drastically different . As far as I can see they both include 2 whole atoms/points in the...

The question simply asks primitive pythagorean triples ##(a,b,c)## such that ##S = a + b + c <15\times 10^{5}## import time import math start = time.perf_counter() pythagorean_triples = {(3, 4, 5) : 12} for m in range(0, 10**3, 2): for n in range(1, m, 2): if math.gcd(m, n) == 1...

Recently I found this online: https://www.html5canvastutorials.com/labs/html5-canvas-graphing-an-equation/ It is a html page using the <canvas> element and it can graph functions. Although much better versions of this is already everywhere(like desmos), it should still be useful considering...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help in order to fully understand the proof of Proposition 4.3.14 ... ... Proposition 4.3.14 reads as follows: In the above proof by Bland we read...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need yet further help in order to fully understand the proof of Proposition 4.3.14 ... ... Proposition 4.3.14 reads as follows: In the above proof by...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some further help in order to fully understand the proof of Proposition 4.3.14 ... ... Proposition 4.3.14 reads as follows: In the above proof by...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help in order to fully understand the proof of Proposition 4.3.14 ... ... Proposition 4.3.14 reads as follows: In the above proof by Bland we...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help to fully understand the proof of Lemma 4.3.10 ... ... Lemma 4.3.10 and its proof read as...

Does anyone know of any resources on questions on primitive roots and order of a modulo n? They need to be suitable for elementary number theory course. (These could be interesting results and challenging ones).

I am teaching elementary number theory to first year undergraduate students. How do introduce the order of an integer modulo n and primitive roots? How do I make this a motivating topic and are there any applications of this area? I am looking at something which will have an impact.

I am teaching elementary number theory to first year undergraduate students. How do introduce the order of an integer modulo n and primitive roots? How do I make this a motivating topic and are there any applications of this area? I am looking at something which will have an impact.

Homework Statement Find the known pythagorean triangles with sides of integers lengths, given the area of the pythagorean triangle is 60. Homework Equations Pythagorean triangles are right angled triangles. a primitive pythagorean triple is of the form: x=2mn y=m^2-n^2 z=m^2+n^2 gcd(m,n)=1...

The problem I am trying to calculate the integral $$ \int_{\gamma} \frac{z}{z^2+4} \ dz $$ Where ## \gamma ## is the line segment from ## z=2+2i ## to ## z=-2-2i ##. The attempt I would like to solve this problem using substitution and a primitive function to ## \frac{1}{u} ##. I am not...

In the first Feynman diagram, an electron comes in, emits a photon and then leaves. Is this an allowed process? Because if you rotate the diagram by 90o, the diagram should be just as valid, but it doesn't seem to be since it would violate the law of conservation of momentum. So is the...

Homework Statement Prove that 2 is a primitive root of ##\mathbb{Z}/83\mathbb{Z}## by hand. Hint: Think hard about ##2^{41}##. Homework Equations Euler's theorem, Euler's Totient function, Chinese remainder theorem(not sure if its relevant). We don't really have anything else. The Attempt at...

Homework Statement I need find the function ##F(x)## . Homework Equations ##\int_0^r F(x)dx = \frac{r^3}{(r^2+A)^{3/2}}+N## where ##A,N## are constants. The Attempt at a Solution I tried using some function of test, for instance the derivative of the right function evaluated in x. But , i...

Lying 620 million light-years away, this galaxy has the lowest oxygen abundance ever seen in a star-forming galaxy. Link: New Scientist

Homework Statement When calculating the Fourier coefficients of the potential of the following lattice (the potential is a sum of deltas at the atom sites): I get the wrong coefficients if I choose the following primitve cell, with primitve vectors a1,a2: And the right coefficients if I...

The problem $$ \int \frac{x}{\sqrt{x^2+2x+10}} \ dx $$ The attempt ## \int \frac{x}{\sqrt{x^2+2x+10}} \ dx = \int \frac{x}{\sqrt{(x+1)^2+9}} \ dx## Is there any smart substitution I can make here to make this a bit easier to solve?

I have to find a primitive function below using the Euler formulas for ##\sin x## and ## \cos x## The problem $$ \int e^{2x} \sin 3x \ dx $$ Relevant equations ## \cos x = \frac{e^{ix}+e^{-ix}}{2} \\ \sin x = \frac{e^{ix}-e^{-ix}}{2i} \\ \\ \int e^{ix} \ dx = \frac{e^{ix}}{i} ## The attempt...

I am trying to find primitives to the rational function below but my answer differs from the answer in the book only slightly and now, I am asking for your help to find the error in my solution. This solution is long since I try to include all the steps in the process. The problem $$ \int...

Hello, I am having trouble with solving the problem below The problem Find all primitive functions to ## f(x) = \frac{1}{\sqrt{a+x^2}} ##. (Translated to English) The attempt I am starting with substituting ## t= \sqrt{a+x^2} \Rightarrow x = \sqrt{t^2 - a} ## in $$ \int \frac{1}{\sqrt{a+x^2}}...

Hello friends from afar. I ran into what I felt to be somewhat of an odd question: Prove that some odd numbers are primitive roots modulo pm for each odd prime p and each positive integer m. It feels dodgy given that any odd number n = p1p2 ⋅⋅⋅ ps cannot be a primitive root of a prime number...

Homework Statement let c be a primitive 16th root of unity. How many subfields M<Q(c) are there such that [M:Q] = 2 Homework EquationsThe Attempt at a Solution I think the only subfield M of Q(c) such that [M:Q] = 2 is Q(c^8). Then M = {a+b(c^8) such that a,b are elements of Q}. I'm thinking...

In case the orientation of a primitive cell is not what I want, is there tools to do a user-supplied 3D rotation to bring the primitive cell to the preferential orientation and output the new coordinates? Thanks,

Experimentalists usually provide chemical unit cell information including full symmetry (space group) information of the crystal together with coordinates of independent atoms. But this cannot be directly used by ab initio packages, which requires either primitive cell or unit cell information...

Homework Statement So I need the find the minimal polynomial of the primitive 15th root of unity. Let's call this minimal polynomial m(x) Homework EquationsThe Attempt at a Solution I know that m(x) is an irreducible factor of x^15 - 1 and also that the degree of m(x) is equal to the Euler...

In Material Science, what is a primitive cell when speaking crystals.

Homework Statement Determine all primitive functions for the function: 2x(x^2+3)^4 2. The attempt at a solution When i expanded i got the primitive to be: 2(x^9/9)+3x^8+18x^6+54x^4+81x^2But this was wrong. I am not sure I have understood the question. Help?

So this is kind of a speculative question but it is something I have long wondered about. I know that if I were to suddenly find myself transported backward in time, say 5000 years, or transported to a primitive Earth like planet, chances are that I would probably die in no time at all. I...

I was reading about Gauss's Lemma here: https://cims.nyu.edu/~kiryl/Algebra/Section_3.10--Polynomials_Over_The_Rational_Field.pdf Unfortunately, I am stuck on Lemma 3.10.1 that concludes that the product of a pair of primitive polynomials is itself primitive. I understand about how there is...

Let ζ5 be e2πi/5. Find a monic polynomial of degree two in K(ζ + ζ−1) So, if E/F is a field extension, with α∈E then K(α) = {f(x)∈F[x] | f(α)=0} and m(x) is the minimal polynomial of α over F such that K(α) = [m(x)] where [m(x)] is the ideal generated by m(x). I was thinking maybe (x- ζ -...

Prove that for any primitive Pythagorean triple (a, b, c), exactly one of a and b must be a multiple of 3, and c cannot be a multiple of 3. My attempt: Let a and b be relatively prime positive integers. If $a\equiv \pm1 \pmod{3}$ and $b\equiv \pm1 \pmod{3}$, $c^2=a^2+b^2\equiv 1+1\equiv 2...

hello i have browsed this forum for topics about this, and i found them very enlightening and helped a lot in terms of finding the length of the primitive vectors , the problem that i have is with the direction of said primitive vectors , while in fcc they are more or less easy to visualize in...

Hi, Given a 3D object in R3 space can we represent it using three basic primitive shapes like Sphere, Cone and Cylinder? Would this claim be valid?

Homework Statement Construct $\mathbb{F}_{16}$ as a quotient of $\mathbb{Z}_2[X]$. How many non-zero elements are primitive in this field? Calculate $|GL2_(\mathbb{F}_16)|$. Homework Equations Primitive Theorem The Attempt at a Solution For the first question, I don't know how to construct...

Homework Statement the distribution function: f(x)= x + 1 when -1 < x ≤ 0 -x + 1 when 0 ≤ x < 1 0 otherwiseHomework Equations The Attempt at a Solution on the first interval i found (1/2)x2 +x + c on the second interval -(1/2)x2 + x + c and when integrating the c's will cancel each...

Homework Statement Given that the primitive basis vectors of a lattice are ##\mathbf{a} = \frac{a}{2}(\mathbf{i}+\mathbf{j})##, ##\mathbf{b} = \frac{a}{2}(\mathbf{j}+\mathbf{k})##, ##\mathbf{c} = \frac{a}{2}(\mathbf{k}+\mathbf{i})##, where ##\mathbf{i}##, ##\mathbf{j}##, and ##\mathbf{k}## are...

Hey! :o Can we enumerate the primitive recursive functions?

Hey! :o According to the book that I'm reading, we can define the $\mu-$recursive functions inductively, as follows: The constant, projection, and successor functions are all $\mu-$recursive. If $g_1, \dots , g_m$ are $n-$variable $\mu-$recursive functions and $h$ is an $m-$variable...

I started by setting $\alpha= e^{2\pi i/3} + \sqrt[3]{2}.$ Then I obtained $f(x) = x^9 - 9x^6 - 27x^3 - 27$ has $\alpha$ as a root. How can I proceed to find the minimal polynomial of $\alpha$ over $\mathbb{Q},$ and identify its other roots?

Hey! :o I have to show that the following functions and sets are primitive recursive: $$f(x,y)=x+y$$ $$f(x,y)=x \cdot y$$ $$sign (x)=\left\{\begin{matrix} 1 &\text{ if } x=0\\ 0 &\text{ if } x>0 \end{matrix}\right. $$ $$x \dot - y=\left\{\begin{matrix} x-y &\text{ if } x \geq y\\...

Homework Statement Following the amount of C1 front, the output Z be 1 . Following an amount of C2 front, the output Z must be 0 . Otherwise, the output remains unchanged. Homework EquationsThe Attempt at a Solution in Attached file. This is the answer but I really do not understand the...

Is it true that $g$ is a primitive root modulo $p$ if and only if $g^{(p-1)/2} \equiv -1 \pmod p$?

Hi Guys, please help me. how can i derive the primitive vector of copper oxide (I)? basically this is cuprous oxide having a cubic crystal structure but since it has oxygen in it the directions and magnitude of primitive vector are far different compare to basic cubic structure. also please help...

Hi Guys, please help me. how can i derive the primitive vector of copper oxide (I)? basically this is cuprous oxide having a cubic crystal structure but since it has oxygen in it the directions and magnitude of primitive vector are far different compare to basic cubic structure. also please help...

Homework Statement CsCl has a BCC unit cell containing 2 atoms, Cs in the origin of the cell and Cl at the centre of the cell. Describe the CsCl unit cell in terms of its primitive cell+basis. Homework Equations R = n1a1 + n2a2 + n3a3 ~ R is the vector which relates one point of the lattice...