### Proving the Non-Existence of Primitive Roots Modulo 954271**Problem Statement:**Prove that there are no primitive roots modulo \( 954271 = 691 \cdot 1381 \) (691 and 1381 are prime). Do this directly (i.e. without referring to the Primitive Root Theorem or any related results).**Detailed Explanation:**To tackle this problem, a direct proof method needs to be applied. This involves showing that no integer can generate all the units in the multiplicative group modulo \( 954271 \). Let's break down the essential steps required for this proof:1. **Structure of the Multiplicative Group:** - The modulus \( 954271 \) is the product of two primes, 691 and 1381. - The prime factorization of \( 954271 \) is important because it helps in understanding the structure of the group of units modulo \( 954271 \). - The multiplicative group modulo \( 954271 \) has order \((691 - 1)(1381 - 1) = 690 \cdot 1380 = 952200\), which is derived from Euler's totient function \( \phi(954271) \).2. **Properties of Primitive Roots:** - A primitive root modulo \( n \) is an integer \( g \) such that the smallest positive power of \( g \) that is congruent to 1 modulo \( n \) is exactly \( \phi(n) \). - In other words, \( g \) must be a generator of the entire multiplicative group of units.3. **Direct Computation:** - Consider any candidate integer \( g \) for being a primitive root modulo \( 954271 \). - Calculate the order of \( g \) modulo 691 and modulo 1381 separately. - Use the Chinese Remainder Theorem to combine these results and determine if \( g \) can generate \( 952200 \) units.4. **Verification:** - For \( g \) to be a primitive root modulo \( 954271 \), its order modulo 691 and 1381 must be \( 690 \) and \( 1380 \) respectively. - Because \( 954271 = 691 \times 1381 \), if there exists a primitive root \( g \) modulo \(

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Answered: Prove that there are no primitive roots…

Prove that there are no primitive roots modulo 954271 = 691 · 1381 (691 and 1381 are prime). Do this directly (i.e. without referring to the Primitive Root Theorem or any related results).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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