Skip to main content
\( \newcommand{\lt}{ < } \newcommand{\gt}{ > } \newcommand{\amp}{ & } \)

Section19.6Exercises

1

Review the proof of Fact 9.5.2 that \(\phi(n)\) is multiplicative. Can you think of a way to modify it directly to prove that \(\sigma\) or \(\sigma_0\) are multiplicative?

My students discovered various facts about the functions in this chapter on their own; why not you?

2

Conjecture and prove a formula for the difference between \(\sigma_k(p)\) and \(\sigma_k(p^2)\). (Thanks to Becca Brule and Olivia Gray.)

3

Conjecture and prove a necessary (or even sufficient) criterion for when \(5\mid \sigma_2(2k)\). (Thanks to Andrew Kwiatkowski and Daniel Brito.)

4

Come up with some new (to you) conjecture about one of these functions you observed from the data, and which isn't mentioned in this book. Tell what led you to this conjecture.

6

Do you think these numbers should be called perfect, and why? Establish a connection to GIMPS.

7

Can you find a number such that \(\sigma(n)=3n\)?

8

Could there be a function \(g(n)\) which is multiplicative, where \(g(2n)=0\), \(g(n)=a_1=1\) if \(n\equiv 1\text{ (mod }8)\), \(g(n)=a_2\) if \(n\equiv 3\text{ (mod }8)\), \(g(n)=a_3\) if \(n\equiv 5\text{ (mod }8)\), and \(g(n)=a_4\) if \(n\equiv 7\text{ (mod }8)\)?

9

Let \(\tau_o(n)\) and \(\sigma_o(n)\) be the same as \(\tau\) and \(\sigma\) but where only odd divisors of \(n\) are considered; let \(\tau_e\) and \(\sigma_e\) be similar for even divisors of \(n\). Evaluate these functions for \(n=1\) to \(12\), and decide whether each of them is multiplicative or not (either proving it, or showing not by counterexample).

10

Use the estimate toward the end of Section 19.3 for \(\sigma\) to find numbers for which \(\sigma(n)>5n\) and \(\sigma(n)>6n\). (Possibly long.)

11

Discover and prove conditions for which \(\tau(n)\) and \(\sigma(n)\) are even and odd numbers.

12

Show that if \(n\) is odd then \(\tau(n)\) and \(\sigma(n)\) have the same parity.

13

For which types of \(n\) is \(\tau(n)=4\)?

14

Prove that if \(n\equiv 7\) (mod \(8\)), then \(8\mid \sigma(n)\).

15

Show that every prime power is deficient.

16

Show that a multiple of an abundant number is abundant.

17

Find a 4-perfect number.

18

Compute “by hand” \(\sigma_{-1}\) for the numbers up to 30. Come up with and prove a criterion for when \(\sigma_{-1}=2\).

19

Find three pseudoperfect numbers less than 100.

20

Find a weird number less than 100.

21

In the proof of Algorithm 19.4.7, confirm that if \(p_n\), \(p_{n-1}\), and \(q_n\) are prime, then the numbers in question are amicable.

22

Prove the first and second facts about the abundancy index in Fact 19.4.11.

23

Find five numbers that must be abundancy outlaws based on the facts (don't just copy from the list).

24

Fill in the details in the proof of Theorem 19.5.2 (that odd perfect numbers need at least three prime divisors, and that \(3\) and \(5\) would need to be the first two if there were exactly three).

25

Read the article linked right after Fact 19.5.5 about Euler and odd perfect numbers, and restate and reprove his criterion in modern notation.

26

There are always more connections. Here are a few exercises about a formula one would have likely never guessed. \begin{equation*}\left(\sum_{d\mid n}\tau(d)\right)^2=\sum_{d\mid n}\tau(d)^3\end{equation*} First, test it out by hand with \(n=6\) and \(n=8\). Then try it with bigger numbers below:

Start a proof by noting that it's clearly true for a prime power \(n=p^e\), for which \(\tau(p^f)=f+1\), and all divisors of \(n\) look like such a power of \(p\).

Continue the proof by examining the proof that \(\sigma_i\) is multiplicative for what can be said about the divisors of \(mn\), and how a sum over divisors \(d\mid mn\) can be a product of two different sums over divisors of \(m\) and \(n\).