9. The Liouville Lambda (A) function is a number theoretic function- similar to T. g. 6 et al. It is defi ned by if n = 1 X(12) %3D itk2t...+k, if n > 1 = pf'p ... pr e N. where n = kiaka Prove that if n is a perfect square, then X(n) = 1.
9. The Liouville Lambda (A) function is a number theoretic function- similar to T. g. 6 et al. It is defi ned by if n = 1 X(12) %3D itk2t...+k, if n > 1 = pf'p ... pr e N. where n = kiaka Prove that if n is a perfect square, then X(n) = 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Question 9
![**Educational Content on Number Theory**
9. **The Liouville Lambda (λ) Function**
The Liouville Lambda function is a number theoretic function defined similarly to functions such as τ, σ, and φ. It is defined as:
\[
\lambda(n) =
\begin{cases}
1 & \text{if } n = 1 \\
(-1)^{k_1 + k_2 + \ldots + k_r} & \text{if } n > 1
\end{cases}
\]
where \( n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r} \in \mathbb{N} \).
*Prove that if \( n \) is a perfect square, then \( \lambda(n) = 1 \).*
10. **Primitive Roots**
Let \( r \) be a primitive root of some \( n \geq 3 \). Prove that:
\[ r^{\frac{\phi(n)}{2}} \equiv -1 \pmod{n} \]
11. **Integer Function Calculations**
Find \( \tau(n) \), \( \sigma(n) \), \( \lambda(n) \), \( \mu(n) \), \( \omega(n) \), and \( \phi(n) \) for the following integers:
- 2250
- 199
- 286936650
- 22!
*(Note: \(\mu(n)\) is defined on page 112. \(\omega(n)\) is defined on page 111.)*
12. **Divisors of a Function**
Let \( p = 17 \) and \( d \) be a divisor of \( \phi(p) \). Determine \( \psi(d) \) for each \( d \). List every element having order \( d \), for all divisors \( d \), of \( \phi(p) \).
13. **Primitive Roots Calculation**
Calculate all the primitive roots of 41 and 26.
14. **Nonexistence of Primitive Roots**
Demonstrate that 21 has no primitive root.
15. **Indices and Congruences**
Let \( r \) be a primitive root of \( n \). If \( \gcd(a, n) = 1 \), then the smallest positive integer \( k \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2Fb0628ad6-b4fa-4dbb-9100-daf394a54b82%2Fee0zrd3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content on Number Theory**
9. **The Liouville Lambda (λ) Function**
The Liouville Lambda function is a number theoretic function defined similarly to functions such as τ, σ, and φ. It is defined as:
\[
\lambda(n) =
\begin{cases}
1 & \text{if } n = 1 \\
(-1)^{k_1 + k_2 + \ldots + k_r} & \text{if } n > 1
\end{cases}
\]
where \( n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r} \in \mathbb{N} \).
*Prove that if \( n \) is a perfect square, then \( \lambda(n) = 1 \).*
10. **Primitive Roots**
Let \( r \) be a primitive root of some \( n \geq 3 \). Prove that:
\[ r^{\frac{\phi(n)}{2}} \equiv -1 \pmod{n} \]
11. **Integer Function Calculations**
Find \( \tau(n) \), \( \sigma(n) \), \( \lambda(n) \), \( \mu(n) \), \( \omega(n) \), and \( \phi(n) \) for the following integers:
- 2250
- 199
- 286936650
- 22!
*(Note: \(\mu(n)\) is defined on page 112. \(\omega(n)\) is defined on page 111.)*
12. **Divisors of a Function**
Let \( p = 17 \) and \( d \) be a divisor of \( \phi(p) \). Determine \( \psi(d) \) for each \( d \). List every element having order \( d \), for all divisors \( d \), of \( \phi(p) \).
13. **Primitive Roots Calculation**
Calculate all the primitive roots of 41 and 26.
14. **Nonexistence of Primitive Roots**
Demonstrate that 21 has no primitive root.
15. **Indices and Congruences**
Let \( r \) be a primitive root of \( n \). If \( \gcd(a, n) = 1 \), then the smallest positive integer \( k \)
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