Primitive Definition and 145 Threads

Definition: A primitive Pythagorean triple is a triple of natural numbers x,y,z s.t. x^2 + y^2 = z^2 and gcd(x,y,z)=1. note: d|gcd(x,y) => d|x and d|y => d^2|x^2 and d^2|y^2 Now z^2 = x^2 + y^2 => d^2|z^2 => d|z Thus, it follows that for any Pythagorean triple x,y,z, we must have...

Let p be a prime. a) If gcd(k,p-1)=1, then 1^k, 2^k,..., (p - 1)^k form a reduced residue system mod p. b) If 1^k, 2^k,..., (p - 1)^k form a reduced residue system mod p, then gcd(k,p-1)=1.[/color] ================================= I proved part a by first showing that each of 1^k...

Find a primitive root modulo 101. What integers mod 101 are 5th powers? 7th powers? -I tested 2. -2 and 5 are the prime factors dividing phi(101)=100 so i calculated 2^50 is not congruent to 1 mod 101 and 2^20 is not congruent to 1 mod 101. -Therefore 2 is a primitive root modulo 101 I guess...

Please prove that if x is quadratic nonResidue modulo 109 and x also cubic nonresidue modulo 109 than x is guaranteed to be primitive root modulo 109 thanks you very much

[SIZE="3"]Suppose that p and q are odd primes and p=2q+1. Suppose that α∈ Z_p^*,α≢±1 mod p. Prove that α is primitive element modulo p if and only if α^q≡-1 mod q.

If H is a nxn primitive, irreducible matrix, is it always true that Hn-1 > 0? That is, every entry in Hn-1 is positive. From my class notes, the definition of H primitive is that there exists some k>0 such that Hk > 0. And a matrix is irreducible if its digraph is strongly connected (that...

Hi all, I've been staring at this question on and off for about a month: Suppose that p is an odd prime, and g and h are primitive roots modulo p. If a is an integer, then there are positive integers s and t such that a \equiv g^s \equiv h^t mod p. Show that s \equiv t mod 2. I feel as...

Give one primitive element for each of the finite field: F2n (here "2" is the subscript) for n=1, n=2,...,n=8

Can a primitive galaxy nova or supernova? Might the former process be involved with star production or the latter with a quasar?

Help in a primitive! Homework Statement Hello guys! Please, I'm really needing help in a primitive... I don't know, maybe it has a simple solution, but I'm tired and blocked on this... Can you give some lights? Here goes the equation: \int\frac{dx}{x^{2}\sqrt{4-x^{2}}} Homework...

Is there an easy (by which I mean an algorithm polynomial in size of input) way to know whether in the multiplicative group of integers mod P (P is a prime), whether an element is a generator or not?

Hi All, I have the following matrix M = M_1 M_2 M_3 ... M_n M is then a product of n matrices. Each of those has dimension 2n by 2n and has the same "look". Consider M_n: this matrix is equal to the identity matrix, 2n by 2n. The only thing different from the identity is that the 3 by 3...

Is there an explicit formula for finding the antiderivative of a function? I was thinking that perhaps it would be the inverse function of the derivative, but I don't know what that would be off the top of my head.

Let f = X^n + a_{n-1}X^{n-1} + \cdots + a_1X + a_o be a polynomial over \mathbb{F}_q for some prime power q such that the least common multiple of the (multiplicative) orders of its roots (in \mathbb{F}_{q^n}) is q^n -1. I would like to show that one of these roots has order q^n -1. (I.e. the...

Hi, Let E = Q(a), where a is a primitive fifth root of unity. Find a basis for E as a vector space over Q, and express a/a-3 in terms of this basis. I can find a basis for E: {1, a, a^2, a^3}, but am not sure how to express a/a-3 in terms of this basis. Any help much appreciated.

Can anyone tell me how to find the exact number of primitive polynomials of degree n over a finite field F_q? I believe the answer is φ(q^n-1)/n, but I cannot find a proof of this. Thanx in advance.

Let r be a primitive root of a prime number p \geq 3. Prove that if p \equiv 1 (mod 4), then -r is also a primitive root of p. I've been told it's quite easy, but I can't see why it's true for the life of me :frown:

If a is a perfect square then a is not a primitive root modulo p (p is an odd prime). (from Artin's conjecture on primitive roots) http://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots This is what I know: suppose a = b^2 a is a primitive root mod p when , a^(p-1) congruent to 1...

Homework Statement Here's a problem I'm having: The primitive vectors of the hexagonal lattice are: a1 = ck a2 = (a/2)i + ([a√3]/2)j a3 = (-a/2)i + ([a√3]/2)j Find the primitive vectors of the reciprocal lattice, i.e. b1, b2, and b3. Homework Equations I do know that the...

Homework Statement Hi guys, i have never taken number theory yet now I am forced to quickly understand it as it was required for a class i signed up. I need help with these problems and would greatly appreciate any hints or help in the right direction. Thanks. 1)Find with proof, all n such...

[SOLVED] primitive cell Homework Statement http://en.wikipedia.org/wiki/Primitive_cell Could someone give me a mathematically rigorous of a primitive cell? I have read like 4 different definitions and none of them seem mathematically rigorous. I think Wikipedia's definition is particularly...

I don't understand this question at all. But since my Professor gave it to me I'm guessing it relates to the chapter of structures and types of solids. What is the coordination number of an atom in a primitive cubic structure? What the hell is this question asking for?

Hi I noticed that multiplication of all primitive roots modulo p ,p>3, congruent to 1 modulo p... I have tried some examples (13,17,19...) but i couldn't prove the general case (let g1...gk be primitive roots modulo p,p>3 ==> g1*g2*...*gk=1(p)) I need help to prove or disprove...

Here is the question: ----------- Suppose n has a primitive root g. For which values of a (in terms of the primitive root g) does the equations x^2 \equiv a \ \text{(mod n)} have solutions? ----------- I really don't have much of an idea of how to even begin this one. Let g be a primitive...

Here is the question from the book: ------------ Determine a primitive root modulo 19, and use it to find all the primitive roots. ------------ \varphi(19)= 18 And 18 is the order of 2 modulo 19, so 2 is a primitive root modulo 19, but I am not sure of how to use that to find all...

If a is a perfect cube, a= n^3, for some integer n, and p is a prime with p is congreunt to 1 mod 3, then show that a cannot be a primitive root mod p, tat is ep(a) is not equal to p - 1

Homework Statement I'm stuck trying to solve a differential equation at the point i need to calculate the primitive of sen x^3 Homework Equations The Attempt at a Solution I've thought on primitives by parts but I don't know how will I do it...

w_{n} is primitive root of unity of order n, w_{m} is primitive root of unity of order m, all primitve roots of unity of order n are roots of Cyclotomic polynomials phi_{n}(x) which is a minimal polynomial of all primitive roots of unity of order n , similarly, phi_{m}(y) is a minimal...

All right, thanks to everybody's help, I've got the algorithm for determining the nth roots of unity. However, for determining which ones are primitive, I'm still having a little trouble. If n is odd, I can test if (n, k) are coprime. However, for even values of n, it seems that it doesn't...

I know that tu solve this primitive we have tu use the substitution method,but I think that none of the rules that should be used apply to this!The problem is, to use the substitution: x=a/b sen t we should have the function in the format (sqrt(a^2 - b^2*x^2)),but instead of a x^2 I have a x^4...

Hello, I got pointed to this forum by OfficeShreder. I have a question I've been puzzling myself over for a while. I am currently trying to implement the "Discrete Weighted Transform". I have reached a step where I need to determine "a primitive Nth root of unity in the appropriate domain"...

I was curious why primitive roots are so important? Also, how one would find out if a number has a primitive root and what and how many of them they are?

I have the definition that if F is a finite field then a \in F is a primitive root if ord(a) = |F|-1. Now what I don't understand is how exactly are there \phi(|F|-1) primitive roots? (Note: This material is supposed not to use any group theory.)

This is the question: http://www.maths.tcd.ie/~cockburd/Question.jpg It's not a homework question, merely one from a previous years tutorial sheet. Okay so it's tetragonal conventional unit cell, has a sort of face centered structure right? Em.. so that would mean it has a four point basis...

How would I go about finding a number that is a primitive root modulo 125? There definitely exists a primitive root since 5^3 =125 The problem basically comes down to finding 'a' (primitive root); a^50 congruent to -1 (mod 125) Anyone know a way apart from trial and error? Thanks

Hello! I know that the primitive variables in fluid mechanics are velocity and pressure. But why? I don't see how "primitive" these variables are...What does it mean by a "primitive variable"? Is that all other variables in a fluid flow can be derived from velocity and pressure? Is there any...

Can a number have more than 1 primitive root? Thanks

Hi, I don't know how to find this primitive function: \int \frac{dx}{(1+\tan x)(1+\tan^2 x)} I tried substitutions t = \tan x or t = 1 + \tan x, but it didn't seem to help me lot... Could someone please point me to the right direction? Thank you.

primitive polynomial in GF(4) ?? Hai All, I m required to make a pseudo random generator. I know that i can make that using some Flip flops and XOR gates(in linear feedback shift register configuration ). But the resulting PN sequences will be in Galois Field(2) as the taps for the flip...

Please any idea on this,find the primitive of arctan x

Let f be a primitive recursive total function, and let A be the set of all n such that the value f(n) is 'new' in the sense of being different from f(m) for all m<n. Show that A is primitive recursive. How in the world do I attack this problem? I am totally lost. Any help would be greatly...

this problem is annoying. I've found that the primitive roots of 9 are 2 and 5. since 2|18 it can't be a root. i know via some theorems in my book that if 5 is a primitive root of 3, then its a primitive root of 3^k, and also of 2*3^k. sorry about not using latex, shouldn't need it for...

In the Davis, Sigal & Weyuker textbook "Computability, Complexity and Languages" their proof that f(x) = x + y goes like this: \begin{equation*}\begin{split}f(x,0) &= u_1^1(x)\\f(x,y+1) &= g(y,f(x,y),x)\end{split}\end{equation*} where g(x_1,x_2,x_3) = s(u_2^3(x_1,x_2,x_3)). And (I'm...

let g be a primitive root of the odd prime p show that -g is a primitive root or not according as p==1 ( mod 4) or p==3(mod4) how would i start in solving this problem thanks :cool:

Hi everyone. I would like to build my own foundry, if possible to melt iron. From all the research that I have done, I can see that almost everyone opts to melt aluminum, even when it comes to building a metal lathe. The reason why I would like to melt iron/steel is that I am also wanting to...