For each k ≥ 1, there exists N such that for every prime p > N, the modular equation xk + yk ≡ zk (mod p) has nonzero solutions x,y,z. For the proof of Schur’s theorem, we will make use of the following fact which says that the multiplicative group of Z/pZ is cyclic. Fact For every prime p, there exists 1 ≤ a ≤ p −1 such that in Z/pZ,

For each k ≥ 1, there exists N such that for every prime p > N, the modular equation xk + yk ≡ zk (mod p) has nonzero solutions x,y,z. For the proof of Schur’s theorem, we will make use of the following fact which says that the multiplicative group of Z/pZ is cyclic. Fact For every prime p, there exists 1 ≤ a ≤ p −1 such that in Z/pZ,

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.2: Properties Of Group Elements

Problem 20E: 20. Let and be elements of a group . Use mathematical induction to prove each of the following...

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