a. Explanation Given information: a . Fermat’s last theorem says that for all integers n > 2 the equation x n + y n = z n has no positive integer solution. Prove the following:If for all prime numbers p > 2 , x p + y p = z p has no positive integer solution, then for any integer n > 2 that is not a power of 2 , x n + y n = z n has no positive integer solution. Calculation: The given information states that to proveif for all prime numbers p > 2 , x p + y p = z p has no positive integer solution, then for any integer n > 2 that is not a power of 2 , x n + y n = z n has no positive integer solution. a divides b if there exists an integer c such that b = a c such that a | b . So, for any integer a and any prime number p , if p | a , then p | ( a + 2 ) . Let us assume that n is an integer such that is not a power of 2 such that n > 2 and such that x n + y n = z n has positive integer solutions. Let a , b and c be positive integers that satisfy x n + y n = z n (which has to exist since x n + y n = z n has positive integer solutions): a n + b n = c n Since n is an integer that is not a power of 2 and since any integer is divisible by a prime number by the uniqueness factorisation of integers theorem, there exists a prime number p such that n is divisible by p b. To determine To prove: n is a power of 2 and n > 4 has no positive integer solution.

Discrete Mathematics: Introduction to Mathematical Reasoning

1st Edition

ISBN: 9780495826170

Author: Susanna S. Epp

Publisher: Cengage Learning

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Chapter 4.6, Problem 31ES

To determine

To prove: The all prime number has no positive integer solution.

To determine

To prove: n is a power of 2 and n>4 has no positive integer solution.

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Chapter 4.6, Problem 30ES

Chapter 4.6, Problem 32ES

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Chapter 4 Solutions

Discrete Mathematics: Introduction to Mathematical Reasoning

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To prove: The all prime number has no positive integer solution.

Solution Summary: The author explains Fermat's last theorem that all prime numbers have no positive integer solution.

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To prove: The all prime number has no positive integer solution.

To prove: n is a power of 2 and n>4 has no positive integer solution.

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Chapter 4 Solutions