. We have seen that for every odd prime p the quadratic congruenceT²-1=0 (mod p)has exactly two solutions x = ±1 (mod p). Show that if p is of the form p = 3k + 2, forsome k € N, then the cubic congruence2³-1=0 (mod p)has the single solution z = 1 (mod p).Hint. What would Fermat do?

Let be a prime of the form for some .Consider the congruence equation .We need to show that it has…

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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Question

. 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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