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Use Theorem 5.2.2 to prove that if a andn are positive integers and d" – 1 is prime, then a = 2 and n is prime. - %3D

Use Theorem 5.2.2 to prove that if a andn are positive integers and d" – 1 is prime, then a = 2 and n is prime. - %3D

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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**Theorem Application and Proof in Number Theory** The problem requires using Theorem 5.2.2 to demonstrate that if \( a \) and \( n \) are positive integers, and \( a^n - 1 \) is prime, then \( a = 2 \) and \( n \) is prime. **Key Points:** - **Positive Integers**: Both \( a \) and \( n \) must be positive integers for this problem. - **Prime Condition**: The expression \( a^n - 1 \) is given as prime. A prime number has exactly two distinct positive divisors: 1 and itself. - **Objective**: Prove that under these conditions, \( a \) must equal 2, and \( n \) must be a prime number. **Suggested Approach:** 1. **Understanding Theorem 5.2.2**: The theorem likely outlines properties related to prime numbers, powers of integers, or specific conditions when exponents are involved. 2. **Exploration**: - Start by assuming \( a = 2 \) and evaluate the expression \( 2^n - 1 \). - Consider the primality of \( n \) based on divisors of the expression. 3. **Analyzing the Expression**: - If \( n \) is not prime, express \( n \) as a product of two integers, say \( p \times q \), and check the implications on \( 2^n - 1 \). 4. **Proof by Contradiction or Example**: Use specific examples or logical contradiction to cement the argument that \( a = 2 \) and \( n \) must be prime. **Conclusion**: By carefully applying Theorem 5.2.2 and analyzing potential integer values, we confirm \( a = 2 \) and \( n \) is prime when \( a^n - 1 \) yields a prime number.
Transcribed Image Text:**Theorem Application and Proof in Number Theory** The problem requires using Theorem 5.2.2 to demonstrate that if \( a \) and \( n \) are positive integers, and \( a^n - 1 \) is prime, then \( a = 2 \) and \( n \) is prime. **Key Points:** - **Positive Integers**: Both \( a \) and \( n \) must be positive integers for this problem. - **Prime Condition**: The expression \( a^n - 1 \) is given as prime. A prime number has exactly two distinct positive divisors: 1 and itself. - **Objective**: Prove that under these conditions, \( a \) must equal 2, and \( n \) must be a prime number. **Suggested Approach:** 1. **Understanding Theorem 5.2.2**: The theorem likely outlines properties related to prime numbers, powers of integers, or specific conditions when exponents are involved. 2. **Exploration**: - Start by assuming \( a = 2 \) and evaluate the expression \( 2^n - 1 \). - Consider the primality of \( n \) based on divisors of the expression. 3. **Analyzing the Expression**: - If \( n \) is not prime, express \( n \) as a product of two integers, say \( p \times q \), and check the implications on \( 2^n - 1 \). 4. **Proof by Contradiction or Example**: Use specific examples or logical contradiction to cement the argument that \( a = 2 \) and \( n \) must be prime. **Conclusion**: By carefully applying Theorem 5.2.2 and analyzing potential integer values, we confirm \( a = 2 \) and \( n \) is prime when \( a^n - 1 \) yields a prime number.
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