4383_Notes_5o1o2

4383_Notes_5o1o2 - Math 4383 Sections 5.1 and 5.2 Page 1 of...

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Math 4383 Sections 5.1 and 5.2 Page 1 of 9 Number Theory Chapter 5 Sections 5.1 and 5.2 – Fermat’s Theorem Pierre de Fermat 1601-1665 Fermat was a lawyer and magistrate in provincial parliament in Toulouse, France. Mathematics was his hobby. Most of his work comes to use through letters he wrote to other mathematicians of his day. Many of the proofs were kept secret His “Last Theorem” – jotted down in the margin of a book he was reading. Section 5.2 – Fermat’s “Little Theorem” Let p be a prime and suppose p does not divide the integer a. Then ( 29 1 1 mod p a - . Proof #1 Consider the set of integers a, 2a, 3a, 4a, 5a, … , (n-1)a

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Math 4383 Sections 5.1 and 5.2 Page 2 of 9 Corollary: Let p be a prime and a be any integer. Then, ( 29 mod p a . Proof: Proof #2 of Fermat’s Theorem Prove: ( 29 mod for all integers a and a prime p. Case 1: a positive
Math 4383 Sections 5.1 and 5.2 Page 3 of 9 Case 2: If a<0 A simple application: Find the remainder when 43 5 is divided by 11.

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This note was uploaded on 02/21/2012 for the course MATH 4383 taught by Professor Flagg during the Spring '09 term at University of Houston.

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4383_Notes_5o1o2 - Math 4383 Sections 5.1 and 5.2 Page 1 of...

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