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Question 5. Let p= 3 (mod 4) be a prime. Assume that a is an integer such that the order of a is (p-1)/2, i.e., a(p-¹)/2 = 1 (mod p) but a" # 1 (mod p) for any n with 1 ≤ n < (p-1)/2. Prove that – a is a primitive root modulo p.

Question 5. Let p= 3 (mod 4) be a prime. Assume that a is an integer such that the order of a is (p-1)/2, i.e., a(p-¹)/2 = 1 (mod p) but a" # 1 (mod p) for any n with 1 ≤ n < (p-1)/2. Prove that – a is a primitive root modulo p.

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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Question 5. Let p = 3 (mod 4) be a prime. Assume that a is an integer such that the order of a is (p-1)/2, i.e., a(p-¹)/2 = 1 (mod p) but aª ‡ 1 (mod p) for any n with 1 ≤ n < (p−1)/2. Prove that –a is a primitive root modulo p.
Transcribed Image Text:Question 5. Let p = 3 (mod 4) be a prime. Assume that a is an integer such that the order of a is (p-1)/2, i.e., a(p-¹)/2 = 1 (mod p) but aª ‡ 1 (mod p) for any n with 1 ≤ n < (p−1)/2. Prove that –a is a primitive root modulo p.
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