Primitive Definition and 145 Threads

Prove that the order of 5 modulo $2^k$ is $2^{k-2}$ where $k$ is at least 3. I thought probably induction is best bet. For k=3 we can verify. So for inductive step we need to show the order of 5 modulo $2^{k+1}$ is $2^{k-1}.

Basically, I've written some code that take as inputs 1)Basis vectors 2)lattice translation vectors and computes the structure factor of the basis, producing a diffraction pattern. I'd like to begin incorporating subtle differences between atoms, so I want to compute the structure factor of...

In Kittel's solid state text, problem 2.3, he says that the volume of the Brillouin zone is the same as a primitive parallelepiped in Fourier space. Somehow I can't see why this is true. Can someone help me see why this is true? Also, is the same relationship true between Wigner-Seitz cells and...

Homework Statement Find the primitive function s(t) for v(t)=3(sin(3t) + cos(3t)) for which s(0)=0 The Attempt at a Solution v(t) =3(sin(3t) + cos(3t)) = 3sin3t +3cos3t s(t) =cos3t -sin3t + C 0 = cos3(0) - sin3(0) + C 0= 1 + C C=-1 such that my primitive function is cos3t -...

Homework Statement In F17, 2 is a primitive 8th root of unity. Evaluate f(x) = 7x3+8x2+3x+5 at the eight powers of 2 in F17. Verify that the method requires at most 16 multiplications in F17. Homework Equations You can can more clearly see the theorem on page 376-378 and the problem is on page...

I've been asked in an exercise to show that the function $f(n)$ which returns the $n$th prime is a primitive recursive function. We've covered the basics of primitive recursion, the primitive recursive schematic notation, addition, multiplication, limited subtraction, bounded products, sums...

Prays the CFT that all function f(x) can be expressed how the integral of its derivative more an initial constant:f(x)=\int_{x_0}^{x}f'(u)du+f(x_0) So, is correct affirm that integral, primitive and antiderivative are concepts differents? ie: f(x) = primitive ##\int_{x_0}^{x}f'(u)du## =...

Hi all, I read The unit cell is the smallest structure that repeats itself by translation through the crystal. Some says premitive unit cells contains atoms only at the corners while a unit cell may contain extra atoms in between(like bcc or fcc). At one place I found this: For each...

Homework Statement The vectors r1 and r2 below represent atomic positions in a crystal. r1 = (n1 + n3)ax + (n2 + n3)ay + n3az r2 = (n1 + n3 + 1/2)ax + (n2 + n3 1/2)ay + (n3 + 1/2)az Assume first that the two vectors above correspond to two different types of atom. Find a set of...

Homework Statement p prime, If p=1 ( mod 3) then Zp contains primitive cube roots of unity. Now I am considering which p does Zp contains primitive fourth roots of unity. opposite way? I mean if p=1(mod4) then Zp contains primitive fourth roots of unity?? 2. The attempt at a solution I...

How can I find a primitive root modulo 169? I found the primitive roots mod 13 by testing 2, and then concluding that any 2^k with (k, 12)=1 would do. So that gave me 2, 6, 7 and 11. But modulo 13 I have no idea how to start.. I’m sure there’s a smarter way than trying 2^the orders that divide...

Homework Statement I was trying to figure out whether or not ##\zeta_5 + \zeta_5^2## and ##\zeta_5^2 + \zeta_5^3## were complex (where ##\zeta_5## is the fifth primitive root of unity). Homework Equations The Attempt at a Solution ##\zeta_5 + \zeta_5^2 = \cos(2\pi/5) + i\sin(2\pi/5) +...

Homework Statement Let ##ζ_3## and ##ζ_5## denote the 3rd and 5th primitive roots of unity respectively. I was wondering if I could write the product of these in the form ##ζ_n^k## for some n and k.Homework Equations The Attempt at a Solution We know that ##ζ_3## is a root of ##x^3=1##, and...

Hi, I am interested in building a very primitive phased array radar for short ranges(1m to 100m). I thought it's possible to use a simple 2.4GHz RF ASK transmitter, and by just sending a single '1' bit I could generate a pulse and then use a simple RF receiver for that frequency and analyze the...

Show that if $x$ is a primitive root of p, and $x^{p-1}$ is not congruent to 1 mod$p^2$, then x is a primitive root of $p^2$

[FONT=CMR12]let p be an odd prime. Show that if there is a primitive root of p^n, then there is a primitive root of 2p^n. Strategy? [FONT=CMR12]

I know that (a^2-b^2,2ab,a^2+b^2) is pythagorean triple. How to show it is primitive? i.e gcd(x,y,z)=1

Homework Statement Find the translation vectors of the primitive cell. The figures are provided. Homework EquationsThe Attempt at a Solution I found these are more difficult than what I am learning from Kittel. I did number 1, but not sure if it is correct. Answer or hint for any of the...

Hi guys. I am old person but not very old. I have a qustion that is very important for me to be answered. How can you cut simple geometrical shapes from a irregular solid using simple methods only? By simple geometrical shapes I mean cuboid, cylinder, pyramid and sphere. stuff like that. You are...

What are the primitive roots of Z_32? \varphi(\varphi(32))=8 However you must first check that there is a primitive root. A PR exists if (a) n=2,4 (b) n=p^k (c)n=2p^k According to the solutions, Z_32 has no primitive roots. Is this correct? 32=2^5 which fulfills one of the conditions (b) so...

Homework Statement I have y'(x)= xcos(x2) How do I get the primitive function for this? Homework Equations The Attempt at a Solution I know that f'(x)=2xcos(x2) is f(x)=sin(x2) How will removing that 2 in front of xcos affect the primitive function?

hi guys..so this is my first post..i would like to start of with what everyone's talking 'bout;yes the 'curiosity'..how did NASA and JPL manage to pull that off! Well...so i got thinking and decided to design a primitive curiosity which moves along certain- coloured paths...awaiting your...

Homework Statement Knowing that sodium has a BCC structure, a density of 970 kg/m^3 and each atom has an atomic mass of 22.98 amu, determine the length of the primitive cell. Homework Equations Not sure. The Attempt at a Solution I calculated how many atoms of Na there is in 1m^3...

I have a question; Let m have a prime factor p \equiv 1 (mod 4). Then Euler function \varphi(m) is divisible by 4. Let x = r^{\varphi(m)}, then m|(x^4-1) and x^4-1=(x^2-1)(x^2+1). As gcd(x^2-1,x^2+1)|2, either x^2-1 or x^2+1 is divisible by m. My book says here because of the nature of a...

Suppose I only have a conventional cell with atomic positions obtained from a structural database, and I need to know the primitive cell for that lattice. Is there some general way to reduce the conventional cell to the primitive cell? That is, to determine the correct primitive vectors and...

Given the domain ℂ\[-1,1] and the function, f(z)=\frac{z}{(z-1)(z+1)}, defined on this domain, the Residue Theorem shows that for \alpha a positive parametrization of the circle of radius two centered at the origin, that: \int_{\alpha}f(z)=\int_{\alpha}\frac{z}{(z-1)(z+1)} = 2\pi i Can I...

Homework Statement given the following position vector: R = (10n1 + 9n2 + 19n3)(a/10) x + 6(n2+n3)(a/5) y + 2(n3)a z where n1, n2 and n3 are integers Find the primitive lattice vectors. Homework Equations any position vector of a lattice point is of the type R= c1 a1 + c2 a2 + c3 a3; and...

Homework Statement Here is a question that I just came across while reading: Which of the following has the most primitive respiratory system? A. Rat B. Fish C. Toad D. Grasshopper E. Lizard I read thoroughly, with a view of having an answer to the question. Uptill now I have not made...

Homework Statement Let f(z) be a complex function analytic everywhere except at a where it has a singularity. Prove that the function f(z) - \frac{b_{-1}}{z-a} has a primitive in a punctured neighborhood of a. Where b_{-1} is the coeffecient of the n=-1 term in the Laurent expansion of...

Gamma is the unit circle oriented counterclockwise. $\int_{\gamma}\frac{e^z}{z}dz$ Instead of finding the power series to solve the integral, I am trying to do it b use of the primitive. However, I seem to not be good at finding the primitives. How can I find the primitive for this?

How can I find the primitive of $\int_{\gamma}ze^{z^2}dz$ from $i$ to $2-i$?

Homework Statement Show that primitive n-th roots of unity have the form e^{i2\pi k/n} for k\in\mathbb{Z},n\in\mathbb{N}, k and n coprime. The attempt at a solution So the n-th roots of unity z have the property z^{n}=1. I have previously shown that (e^{2\pi ik/n})^{n}=e^{2\pi ik}=(e^{2\pi...

Hello! I am having trouble understanding the whole primitive function thing in complex analysis. How do I check if a certain function, say [itex] \frac {sin(z)} {z} [/tex] has a primitive, say in C/{0} ? What is the intuition? My intuition is that 0 is the only singularity, so if it is...

Let's recall the Euclidean Rule for Pythagorean Triangles: Let (m,n) be co-prime natural numbers (m<n), then h := n^{2} + m^{2} e := 2 m n d := n^{2} - m^{2} = (n - m) * (n + m) form the hypothenuse, the even and the odd leg of a primitive Pythagorean triangle (PPT) If we...

How to prove there is no primitive cell of diamond structure with only one atom? Thanks in advance!

How can i develop a sketch of the lattice and reciprocal lattice from vector form a=i+4j b=3i i know how to draw the wigner site cell, but I am having problems developing a sketch from vectors. what is the method for working it out..please help

Hi to all experts! I know individually about primitive cell("It has lattice point at corners only") and unit cell("It has lattice point at corners as well as at center if bcc or at faces if fcc or at bases if it is base centered") Are these right? But I don't know what is Wigner-Seitz cell...

Homework Statement if p is a prime of the form p=4k+1 and g is a primitive root of p, show that -g is a primitive root. I'm not sure if this is a decent proof or not. My final argument looks suspicious. Any thoughts? Thanks Tal The Attempt at a Solution First, notive that...

Homework Statement Hi, I need to show that \alpha+1=[x] is a primitive element of GF(9)= \mathbb{Z}_3[x]/<x^{2}+x+2> I have already worked out that the function in the < > is irreducible but I do not know where to go from this. Homework Equations there are 8 elements in the...

Hi people Can you kindly tell me how to solve P(cotg x)^5 step by step? I made some calculation but I couldn't achieve any result Thank you :)

Hi We know that there is no unique primitive cell, meaning there is no unique choice of primitive vectors. Now, when we find our reciprocal primitive vectors, then we can construct the first Brillouin zone (FBZ) by using the Wigner-Seitz method. But we know that primitive vectors are not...

Homework Statement show that cos(2pi/n) + isin(2pi/n) is a primitive root of unity Homework Equations The Attempt at a Solution if i know z = cos(2pi/n) + isin(2pi/n) is an nth root and I'm trying to prove that z is a primitive nth root. is it correct to assume that z^k is not...

Homework Statement I must show that cos(2pi/n) + isin(2pi/n) is a primitive root of unity Homework Equations a primitive root of unity is an nth root of unity that does not equal 1 when raised to the kth power for k less than n and great than or equal to 1 The Attempt at a Solution...

Im not sure if this should go in the math/number theory section or here, but here it goes: how do programs calculate the primitive roots mod n of extremely large primes? My program will only go up to 12-14 bits before having memory errors caused by storage of the totient of the prime number...

Homework Statement (a) Find a primitive root β of F3[x]/(x^2 + 1). (b) Find the minimal polynomial p(x) of β in F3[x]. (c) Show that F3[x]/(x^2 + 1) is isomorphic to F3[x]/(p(x)). The Attempt at a Solution I am completely lost on this one :confused:

the title says everything. why they don't define set? is it possible to do so? if not, why?

what is exact reason behind choosing basic primitive shapes for elements in FEA? Why it can't be hexagonal, pentagonal, octagonal, etc. in both cases 2D and 3D... Thanks in advance...

designing a two-bit adder using verilog's primitive gates_I get "z" for my outputs Hi, I am trying to write verilog code for a two bit adder using verilog's primitive gates. I had implemented a two bit adder before...using gates I myself had implemented with nmos and pmos transistors, with...

Homework Statement Let n be a nonzero integer and let a be an integer with gcd(a, n) = 1. We have two equivalent conditions which characterize primitive roots: (i) a is primitive modulo n if the order of a is \phi(n). (ii) a is primitive modulo n if for every element b with gcd(b, n) = 1 we...

Theorem: The positive primitive solutions of x^2 + y^2 = z^2 with y even are x = r^2 - s^2, y = 2rs, z = r^2 + s^2, where r and s are arbitrary integers of opposite parity with r>s>0 and gcd(r,s)=1. Using this theorem, find all solutions of the equation x^2 + y^2 = 2z^2 (hint: write the...