Suppose that a and y are integers that satisfyy = a² +2a) By considering congruences modulo 4 prove that r is odd.b) Prove that r+ v-2 and a – V-2 are relatively prime in Z[v-2].c) Deduce that r+v-2 is equal to a cube of a member of Z[V-2]. Show that (r, y) =(±5, 3)%3D

Answer for sub question a: Note: Cube of any integer is congruent to either 0 or 1 or 3 modulo 4.…

Suppose that r and y are integers that satisfy y = x² + 2 a) By considering congruences modulo 4 prove that r is odd. b) Prove that z+ V-2 and r – /-2 are relatively prime in Z[V-2]. c) Deduce that r +v-2 is equal to a cube of a member of Z[/-2]. Show that (r, y) = (±5,3)

Suppose that r and y are integers that satisfy y = x² + 2 a) By considering congruences modulo 4 prove that r is odd. b) Prove that z+ V-2 and r – /-2 are relatively prime in Z[V-2]. c) Deduce that r +v-2 is equal to a cube of a member of Z[/-2]. Show that (r, y) = (±5,3)

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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