Prove that 2 is not a primitive root modulo 31.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question

Transcribed Image Text:I'm unable to transcribe the text that's obscured in blue. However, for educational purposes, let's address the visible text:
**Title:** Proving 2 is Not a Primitive Root Modulo 31
**Introduction:**
In number theory, a primitive root modulo \( n \) is any number \( g \) such that every number coprime to \( n \) is congruent to a power of \( g \) modulo \( n \). More formally, \( g \) is a primitive root modulo \( n \) if the powers of \( g \) generate all the numbers from 1 to \( n-1 \) that are coprime to \( n \).
**Proof:**
To prove that 2 is not a primitive root modulo 31, we need to show that \( 2^k \equiv 1 \pmod{31} \) for some \( k \) that divides \(\phi(31)\), where \(\phi\) is the Euler’s totient function.
Since 31 is prime, \(\phi(31) = 31 - 1 = 30\). Thus, 2 must generate the integers {1, 2, ..., 30} modulo 31.
1. Let's calculate some powers of 2 modulo 31:
- \( 2^1 \equiv 2 \pmod{31} \)
- \( 2^2 \equiv 4 \pmod{31} \)
- \( 2^3 \equiv 8 \pmod{31} \)
- ...
- Calculate up until you notice repeated results or identify \( 2^{15} \equiv 1 \pmod{31} \).
2. Since \( 2^{15} \equiv 1 \pmod{31} \) and 15 is a divisor of 30, 2 is not a primitive root modulo 31 as it does not have an order of 30.
**Conclusion:**
This sequence demonstrates that 2 does not satisfy the requirements of a primitive root since it fails to generate all integers from 1 to 30 before repeating a cycle. Therefore, 2 is not a primitive root modulo 31.
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images

Recommended textbooks for you

Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated

Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY

Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated

Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY

Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,

