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Find the four solutions of x* = 1 (mod 13).

Find the four solutions of x* = 1 (mod 13).

Advanced Engineering Mathematics
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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## Number Theory Exercises ### 1. Find the Four Solutions **Problem Statement:** Find the four solutions of \( x^4 \equiv 1 \pmod{13} \). ### 2. Number Theory Solution **Problem Statement:** Suppose \( p \) is prime and \( d \) divides \( p-1 \). Show that \( x^d - 1 \equiv 0 \pmod{p} \) has exactly \( (d, p-1) \) solutions. Conclude that if \( 4 \mid p-1 \), then \( x^2 \equiv -1 \pmod{p} \) has two solutions, and if \( 4 \nmid p-1 \), then \( x^2 \equiv -1 \pmod{p} \) has no solutions.
Transcribed Image Text:## Number Theory Exercises ### 1. Find the Four Solutions **Problem Statement:** Find the four solutions of \( x^4 \equiv 1 \pmod{13} \). ### 2. Number Theory Solution **Problem Statement:** Suppose \( p \) is prime and \( d \) divides \( p-1 \). Show that \( x^d - 1 \equiv 0 \pmod{p} \) has exactly \( (d, p-1) \) solutions. Conclude that if \( 4 \mid p-1 \), then \( x^2 \equiv -1 \pmod{p} \) has two solutions, and if \( 4 \nmid p-1 \), then \( x^2 \equiv -1 \pmod{p} \) has no solutions.
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