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Chapter 8: Problem 28

For which positive integers \(n\) is \(\phi(n)\) a power of \(2 ?\)

Short Answer

Expert verified
The only positive integers \(n\) for which the \(\phi(n)\) is a power of 2 are those of the form \(2^{m}\) and \(3^{m}\) for \(m \geq 1\).

Step by step solution

01

Describing Euler's totient function \(\phi(n)\) properties

Euler's totient function can be expressed as \(\phi(n) = n(1 - \frac{1}{p_{1}})(1 - \frac{1}{p_{2}})...(1 - \frac{1}{p_{k}})\) where \(p_{1}, p_{2},..., p_{k}\) are distinct prime factors of \(n\). When \(\phi(n)\) yields a power of 2, it means \(n\) should only have single prime factors - such as 2, or \(p^{m}\) wherein \(p\) is a prime number and \(m\) is positive integer.
02

Considering the case of \(n\) being power of 2

First consider the case when \(n\) is a power of 2. \(n = 2^{m}\) leads to \(\phi(n) = 2^{m} - 2^{m-1}\) which simplifies to \(2^{m-1}\). This clearly is a power of 2 for any positive integer \(m\). Therefore, all powers of 2 do satisfy the condition.
03

Considering the case of \(n\) as power of an odd prime

Now let's consider the case of \(n\) equal to an odd prime power \(p^{m}\). For \(p > 2\), we have \(\phi(n) = p^{m} - p^{m-1}\) which simplifies to \(p^{m-1}*(p-1)\). For \(\phi(n)\) to be a power of 2, \(p-1\) must also be a power of 2. The only prime number for which \(p-1\) is a power of 2 is \(p = 3\). Thus, for \(n = 3^{m}\), \(\phi(n)\) is a power of 2.
04

Considering the case of \(n\) having multiple prime factors

If \(n\) is a composite number having multiple distinct prime factors, then from Euler's function definition it follows \(\phi(n)\) would be a product of multiple terms, each less than the corresponding prime. Therefore, \(\phi(n)\) would not be a power of 2.

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Most popular questions from this chapter

At an upcoming family reunion, five families - each consisting of a husband, wife, and one child - are to be seated around a circular table. In how many ways can these 15 people be arranged around the table so that no family is seated all together? (Here, as in Example 8.9, two seating arrangements are considered the same if one is a rotation of the other.)

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