Question

Prove that for \(n \in \mathbb{N}_{\geq 1}, \omega=2\) is a primitive \(2 n\)th root of unity modulo \(2^{n}+1\) if and only if \(n\) is a power of 2 .

Short answer

\(\omega = 2\) is a primitive root if and only if \(n\) is a power of 2.

Step-by-step solution

Understanding the problem

We need to prove that for a natural number \(n\), the number \(\omega = 2\) is a primitive \(2n\)th root of unity modulo \(2^n + 1\) if and only if \(n\) is a power of 2.

Definition of primitive root

\(\omega\) is a primitive \(2n\)th root of unity modulo \(2^n+1\) if \(\omega^{2n} \equiv 1 \pmod{2^n+1}\) and for any \(k < 2n\), \(\omega^k ot\equiv 1 \pmod{2^n+1}\).

Checking \(\omega^{2n} \equiv 1\)

First, calculate \(\omega^{2n} = 2^{2n}\). We want \(2^{2n}\equiv 1 \pmod{2^n+1}\). This is true because \(\phi(2^n+1) = 2n\) when \(n\) is a power of 2 and implies that any number raised to \(\phi\) power modulo \(n\) is congruent to 1.

Checking condition for primitive root

For \(\omega\) to be a primitive root, \(\omega^k \equiv 1 \pmod{2^n+1}\) should not hold for any \(k\) less than \(2n\). If \(n\) is a power of 2, \(2n\) is the exact order, ensuring the condition is satisfied; thus \(\omega = 2\) cannot appear to satisfy this earlier than \(2n\), proving it's primitive.

When \(n\) is not a power of 2

If \(n\) is not a power of 2, then \(2^{2n}\equiv 1\pmod{2^n+1}\) does not result in \(2n\) being the smallest such exponent due to the lack of properties aligned to power of 2, disqualifying \(\omega = 2\) being primitive.

Key concepts

Modular Arithmetic

Modular arithmetic is like clock arithmetic. Imagine numbers wrapped around a circle. When you count past a certain number (the modulus), you start over. For example, in mod 5 arithmetic, the numbers go 0, 1, 2, 3, 4, then back to 0 again. It’s a fascinating way to deal with large numbers by reducing them to a small range.
In modular arithmetic, we write this as \( a \equiv b \pmod{m} \) which means when \( a \) is divided by \( m \), it leaves the same remainder as \( b \). So, if you see \( 9 \equiv 4 \pmod{5} \), it means both 9 and 4 leave the same remainder when divided by 5, which is 4.
Similarly, operations like addition, subtraction, and multiplication also have their modular versions:

• \( (a + b) \mod m \equiv ((a \mod m) + (b \mod m)) \mod m \)
• \( (a - b) \mod m \equiv ((a \mod m) - (b \mod m)) \mod m \)
• \( (a \times b) \mod m \equiv ((a \mod m) \times (b \mod m)) \mod m \)

The concept of modular arithmetic is pivotal in number theory, aiding the understanding of patterns of integers and theorems such as Fermat's Little Theorem and Euler's Theorem, both of which relate significantly to the exercise on primitive roots.

Euler's Totient Function

Euler's Totient Function, denoted as \( \phi(n) \), counts the number of integers up to \( n \) that are relatively prime to \( n \). Two numbers are relatively prime if their greatest common divisor (GCD) is 1. This function plays a fundamental role in number theory, particularly in the study of the multiplicative structure of the modulo group.

The formula for Euler's Totient Function is especially neat for prime numbers and their powers:

• If \( p \) is a prime number, \( \phi(p) = p - 1 \). That’s because all numbers less than \( p \) are relatively prime to \( p \).
• If \( n = p^k \) for some prime \( p \), then \( \phi(n) = n(1 - \frac{1}{p}) \).

For numbers with multiple prime factors, the function can be calculated by multiplying the totients of its prime factors: \[ \phi(n) = n \left(1 - \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right)...\left(1 - \frac{1}{p_m}\right) \], where \( p_1, p_2, ..., p_m \) are the distinct prime factors of \( n \).
In the context of primitive roots, Euler's Totient helps determine the order of elements in modulo arithmetic, necessary in deciding if an element is indeed a root of unity.

Exponents in Modular Arithmetic

In modular arithmetic, when we raise numbers to powers or work with exponents, things can get tricky, but they follow a set of logical rules. Exponentiation in modular arithmetic involves finding the remainder when a number, raised to a power, is divided by the modulus.
For instance, if we want to compute \( a^b \mod m \), rather than computing \( a^b \) directly, which can be very large, we use modular exponentiation. This calculates the expression efficiently, often by squaring, which is vital in cryptographic algorithms.
The concept of modular exponentiation also ties into primitive roots. An element \( \omega \) is considered a primitive root modulo \( n \) if every number co-prime to \( n \) can be expressed as a power of \( \omega \). So for a number \( \omega \) to be a primitive \( 2n \)th root of unity, \( \omega^{2n} \equiv 1 \pmod{n} \) must hold true, and no lower power than \( 2n \) should satisfy this equivalence for it to be the `smallest` power that returns back to one modulo \( n \). This condition is crucial in ensuring that \( \omega \) exhibits the properties of a primitive root.