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Chapter 9: Problem 7

If \(p=2^{k}+1\) is prime, verify that every quadratic nonresidue of \(p\) is a primitive root of \(p\). [Hint: Apply Euler's criterion.]

Short Answer

Expert verified
Every quadratic nonresidue of \( p \) has order \( p-1 \), so it is a primitive root.

Step by step solution

01

Understand Euler's Criterion

Euler's criterion states that for a prime number \( p \), an integer \( a \) is a quadratic residue modulo \( p \) if and only if \( a^{\frac{p-1}{2}} \equiv 1 \pmod{p} \). Conversely, if \( a^{\frac{p-1}{2}} \equiv -1 \pmod{p} \), then \( a \) is a quadratic nonresidue.
02

Determine the Order of a Primitive Root

A primitive root modulo \( p \) is an integer whose order is \( p-1 \). This means that if \( g \) is a primitive root of \( p \), then \( g^k ot\equiv 1 \pmod{p} \) for any \( k < p-1 \).
03

Analyze the Given Prime Number

Given that \( p = 2^k + 1 \), \( p - 1 = 2^k \). The order of any element modulo \( p \) could be a divisor of \( 2^k \), meaning it could be \(2^i\) for \( i = 0, 1, \ldots, k \).
04

Check Terms of Euler's Criterion for Quadratic Nonresidues

For a quadratic nonresidue \( a \), Euler's criterion gives \( a^{\frac{p-1}{2}} \equiv -1 \pmod{p} \). This indicates \( a^{2^{k-1}} \equiv -1 \pmod{p} \).
05

Verify if Quadratic Nonresidue is a Primitive Root

Since \( a^{2^{k-1}} \equiv -1 \pmod{p} \) and \( (a^{2^{k-1}})^{2} \equiv a^{{2^{k}}}\equiv 1 \pmod{p} \), the smallest power where \( a^m \equiv 1 \pmod{p} \) is \( m = 2^k = p-1 \). So, any quadratic nonresidue \( a \) must have order \( p-1 \), proving it is a primitive root.

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Key Concepts

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

Euler's Criterion
Euler's Criterion provides a useful method to determine whether a number is a quadratic residue modulo a prime number, or not. In simple terms, if you have a prime number \( p \) and an integer \( a \), Euler's Criterion tells us:
  • If \( a^{\frac{p-1}{2}} \equiv 1 \pmod{p} \), then \( a \) is a quadratic residue modulo \( p \).
  • If \( a^{\frac{p-1}{2}} \equiv -1 \pmod{p} \), then \( a \) is a quadratic nonresidue modulo \( p \).
Euler's Criterion is vital in number theory since it helps in understanding the properties of numbers in modular arithmetic, especially when dealing with problems involving roots.
Primitive Roots
A primitive root modulo a prime \( p \) is a number that can generate all the non-zero residues modulo \( p \) through its powers. For a number \( g \) to be a primitive root modulo \( p \), its order must be exactly \( p-1 \). This means:
  • \( g^k ot\equiv 1 \pmod{p} \) for any \( k < p-1 \).
  • \( g^{p-1} \equiv 1 \pmod{p} \).
In essence, primitive roots are pivotal because they help in simplifying computations in fields that require full cycles of residue classes, such as cryptography and coding theory.
Prime Numbers
Prime numbers are the building blocks of number theory. A prime number \( p \) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. Some key characteristics of prime numbers include:
  • They have exactly two distinct positive divisors: 1 and themselves.
  • Any number greater than 1 that is not prime is called composite.
Prime numbers are crucial in mathematics because they are used to prove various propositions and theorems, especially in topics involving divisibility and modular arithmetic.
Modulo Arithmetic
Modulo arithmetic, often called "clock arithmetic," is a system of arithmetic for integers, where numbers wrap around upon reaching a certain value, the modulus. It's given by:
  • \( a \equiv b \pmod{m} \) means \( m \mid (a-b) \).
  • The modulus \( m \) divides the difference of \( a \) and \( b \).
Modulo arithmetic is prevalent in fields that require cyclic structures, such as cryptography, computer science, and algebra. It is fundamental because it manages calculations involving cycles, helping to simplify problems into more manageable computations.

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Most popular questions from this chapter

If \(n>2\) and \(\operatorname{gcd}(a, n)=1\), then \(a\) is called a quadratic residue of \(n\) whenever there exists an integer \(x\) such that \(x^{2} \equiv a(\bmod n)\). Prove that if \(a\) is a quadratic residue of \(n>2\), then \(a^{\phi(n) / 2} \equiv 1(\bmod n)\)

(a) Prove that if \(p>3\) is an odd prime, then $$ (-3 / p)=\left\\{\begin{aligned} 1 & \text { if } p \equiv 1(\bmod 6) \\ -1 & \text { if } p \equiv 5(\bmod 6) \end{aligned}\right. $$ (b) Using part (a), show that there are infinitely many primes of the form \(6 k+1\). [Hint: Assume that \(p_{1}, p_{2}, \ldots, p_{r}\) are all the primes of the form \(6 k+1\) and consider the integer \(\left.N=\left(2 p_{1} p_{2} \cdots p_{r}\right)^{2}+3 .\right]\)

Let \(p\) be an odd prime and \(\operatorname{gcd}(a, p)=\operatorname{gcd}(b, p)=1\). Prove that either all three of the quadratic congruences $$ x^{2} \equiv a(\bmod p) \quad x^{2} \equiv b(\bmod p) \quad x^{2} \equiv a b(\bmod p) $$ are solvable or exactly one of them admits a solution.

Solve the following quadratic congruences: (a) \(x^{2}+7 x+10 \equiv 0(\bmod 11)\). (b) \(3 x^{2}+9 x+7 \equiv 0(\bmod 13)\). (c) \(5 x^{2}+6 x+1 \equiv 0(\bmod 23)\).

Prove that there exist infinitely many primes of the form \(8 k+3\). [Hint: Assume that there are only finitely many primes of the form \(8 k+3\), say \(p_{1}\), \(p_{2}, \ldots, p_{r}\), and consider the integer \(N=\left(p_{1} p_{2} \cdots p_{r}\right)^{2}+2\).]
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