We have seen that for every odd prime p the quadratic congruence ²-1=0 (mod p) has exactly two solutions x = ±1 (mod p). Show that if p is of the form p = 3k + 2, fo some k E N, then the cubic congruence 2³-1=0 (mod p) has the single solution z = 1 (mod p).
We have seen that for every odd prime p the quadratic congruence ²-1=0 (mod p) has exactly two solutions x = ±1 (mod p). Show that if p is of the form p = 3k + 2, fo some k E N, then the cubic congruence 2³-1=0 (mod p) has the single solution z = 1 (mod p).
Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter2: Working With Real Numbers
Section2.7: Problem Solving: Consecutive Integers
Problem 14OE
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![. We have seen that for every odd prime p the quadratic congruence
T²-1=0 (mod p)
has exactly two solutions x = ±1 (mod p). Show that if p is of the form p = 3k + 2, for
some k € N, then the cubic congruence
2³-1=0 (mod p)
has the single solution z = 1 (mod p).
Hint. What would Fermat do?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feabade52-5174-4cc3-8205-365538407c90%2Fb9d7f98c-83af-429c-9ebe-ff0d5403d89b%2Fv4gdq79_processed.png&w=3840&q=75)
Transcribed Image Text:. We have seen that for every odd prime p the quadratic congruence
T²-1=0 (mod p)
has exactly two solutions x = ±1 (mod p). Show that if p is of the form p = 3k + 2, for
some k € N, then the cubic congruence
2³-1=0 (mod p)
has the single solution z = 1 (mod p).
Hint. What would Fermat do?
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