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Chapter 4: Problem 2

Compute all primitive roots for \(p=11,13,17\), and \(19 .\)

Short Answer

Expert verified
Primitive roots for \(p=11\) are \(2, 6, 7, 8\); for \(p=13\) are \(2, 6, 7, 11\); for \(p=17\) are \(3, 5, 6, 7, 10, 11, 12, 14\); for \(p=19\) are \(2, 3, 10, 13, 14, 15\).

Step by step solution

01

Understand Primitive Roots

A primitive root modulo a prime number \(p\) is any number \(g\) such that \(g\) raised to the powers from \(1\) to \(p-1\) results in all the numbers from \(1\) to \(p-1\) (mod \(p\)) without repetition. Mathematically, for a prime number \(p\), a number \(g\) is a primitive root mod \(p\) if the smallest integer \(k\) such that \(g^k otot\backslash otin ot\backslash ot= 1\) (mod \(p\)) is \(k=p-1\).
02

List Out Candidates

List out all integers from \(1\) to \(p-1\) for each \(p\). These are candidates for being primitive roots.
03

Check Each Candidate

To determine if a number \(g\) is a primitive root modulo \(p\), compute the powers \(g^1, g^2, ..., g^{p-1}\) (mod \(p\)). If these powers yield all distinct values from \(1\) to \(p-1\) (mod \(p\)) exactly once, then \(g\) is a primitive root modulo \(p\).
04

Apply Steps for \(p=11\)

For \(p=11\), check candidates: \(2, 3, 4, 5, 6, 7, 8, 9, 10\). The primitive roots modulo \(11\) are \(2, 6, 7, 8\).
05

Apply Steps for \(p=13\)

For \(p=13\), check candidates: \(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\). The primitive roots modulo \(13\) are \(2, 6, 7, 11\).
06

Apply Steps for \(p=17\)

For \(p=17\), check candidates: \(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16\). The primitive roots modulo \(17\) are \(3, 5, 6, 7, 10, 11, 12, 14\).
07

Apply Steps for \(p=19\)

For \(p=19\), check candidates: \(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18\). The primitive roots modulo \(19\) are \(2, 3, 10, 13, 14, 15\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Number Theory
Number theory is the branch of mathematics that deals with the properties and relationships of numbers, particularly the integers. Key aspects include:

鈥� Prime numbers
Prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves. They are the building blocks of number theory because every integer can be uniquely factored into prime numbers.

鈥� Divisibility
Divisibility rules help determine if one number is divisible by another. This is important for understanding the structure of integers and for identifying prime numbers.

鈥� Greatest Common Divisor (GCD) and Least Common Multiple (LCM)
The GCD of two numbers is the largest number that divides both of them, while the LCM is the smallest number that is a multiple of both. These concepts are crucial in solving problems involving fractions and modular arithmetic.

鈥� Congruences
In number theory, congruences express that two numbers leave the same remainder when divided by a given number (modulus). This is a fundamental concept in modular arithmetic.
Modulo Arithmetic
Modulo arithmetic, also known as clock arithmetic, deals with the remainders of division operations. It is essential for understanding primitive roots.

鈥� Basics of Modulo Arithmetic
鈥� Operations
鈥� Finding Multiplicative Inverses
Prime Numbers
Prime numbers play a pivotal role in topics like primitive roots and cyclic groups.

鈥� Properties
鈥� Role in Primitive Roots
鈥� Generating Large Prime Numbers
Cyclic Groups
A cyclic group is a mathematical concept where elements can be generated by repeatedly applying a group operation to an initial element (called a generator or primitive root in the context of numbers).

鈥� Definition
鈥� Cyclic Groups in Modular Arithmetic
鈥� Application in Finding Primitive Roots

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Most popular questions from this chapter

If \(p=2^{n}+1\) is a Fermat prime, show that 3 is a primitive root modulo \(p\).

Solve the congruence \(1+x+x^{2}+\cdots+x^{6} \equiv 0(29)\).

Let \(A\) be a finite abelian group and \(a, b \in A\) elements of order \(m\) and \(n\), respectively. If \((m, n)=1\). prove that \(a b\) has order \(m n\).

Determine the numbers \(a\) such that \(x^{3} \equiv a(p)\) is solvable for \(p=7,11\), and 13 .

Calculate the solutions to \(x^{3} \equiv 1(19)\) and \(x^{4} \equiv 1\) (17).
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