- Review the proof from earlier that \(\phi(n)\) is multiplicative. Can you think of a way to modify it to prove that \(\sigma\) or \(\sigma_0\) are multiplicative?
- Conjecture and prove a formula for the difference between \(\sigma_k(p)\) and \(\sigma_k(p^2)\). (Thanks to Becca Brule and Olivia Gray)
- Conjecture and prove a necessary (or even sufficient) criterion for when \(5\mid \sigma_2(2k)\). (Thanks to Andrew Kwiatkowski and Daniel Brito)
- A perfect number is an \(n\) such that \(\sigma(n)=2n\). Read Euclid's proof that certain even numbers are perfect and write it down in modern notation.
- Why are these numbers called perfect, anyway? What does this all have to do with GIMPS?
- Can you find a number such that \(\sigma(n)=3n\)?
- 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 \(n\equiv 7\text{ mod }(8)\)?
- 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).
- Use the estimate in the text for \(\sigma\) to find numbers for which \(\sigma(n)>5n\) and \(\sigma(n)>6n\).
- Discover and prove when \(\tau(n)\) and \(\sigma(n)\) are even and odd numbers.
- Show that if \(n\) is odd then \(\tau(n)\) and \(\sigma(n)\) have the same parity.
- For which types of \(n\) is \(\tau(n)=4\)?
- Prove that if \(n\equiv 7\) mod (8), then \(8\mid \sigma(n)\).
- Show that every prime power is deficient.
- Show that a multiple of an abundant number is abundant.
- Find a 4-perfect number.
- Compute “by hand” \(\sigma_{-1}\) for the numbers up to 30. Come up with and prove a criterion for when \(\sigma_{-1}=2\).
- Find three pseudoperfect numbers less than 100.
- Find a weird number less than 100.
- Confirm that if \(p_n\), \(p_{n-1}\), and \(q_n\) are prime, then the numbers in question are amicable.
- Prove the first and second facts about the abundancy index.
- Here are a few exercises about a formula one would have likely never guessed. \[\left(\sum_{d\mid n}\tau(d)\right)^2=\sum_{d\mid n}\tau(d)^3\]
- Test it out by hand with \(n=6\) and \(n=8\). 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 \(n\).
- Find five numbers that must be abundancy outlaws based on the facts (don't just copy from the list).
- Fill in the details in the proof of odd perfect numbers needing at least three prime divisors and that \(3\) and \(5\) would need to be the first two if that were the case.
- Read the article about Euler and odd perfect numbers, and restate and reprove his criterion in modern notation.
