   Chapter 7.1, Problem 53ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Each of exercises 51-53 refers to the Euler phi function, denoted ϕ , which is defined as follows: For each integer n ≥ 1 , ϕ ( n ) is the number of positive integers less than or equal to n that have no common factors with n except ± 1 . For example, ϕ ( 10 ) = 4 because there are four positive integers less than or equal to 10 that have no common factor with 10 except ± 1 -namely, 1,3,7, and9.Prove that there are infinitely many integers n for which ϕ ( n ) is a perfect square.

To determine

To Prove:

There are infinitely many integers n for which ϕ(n) is a perfect square.

Explanation

Given information:

When p is a prime number and n is an integer with n1, then

ϕ(pn)=pnpn1.

Proof:

The objective is to prove that there are infinitely many integers n for which ϕ(n) is perfect square.

If p is prime then,

ϕ(pn)=pn(11p)     .............(1)

Put p=2 (Since 2 is a prime number) in equation (1), to obtain.

ϕ(2n)=2n(112)=2n12=2n1=( 2 n1 2

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