Number Theory: In Context and Interactive

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?

Read Euclid’s original proof¹⁸ that certain even numbers are perfect and write it down in modern notation.

Do you think perfect numbers as defined in Definition 19.4.1 should be called perfect? Why or why not? Establish a connection to GIMPS.

Please find a number such that \(\sigma(n)=3n\text{.}\) (This was apparently first done in Robert Recorde’s Whetstone of Witte in 1557, where we also find the equals sign for the first time.)

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

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 \(n\text{.}\) Evaluate these functions for \(n=1\) to \(12\text{,}\) and decide whether each of them is multiplicative or not (either proving it, or showing not by counterexample).

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

Discover and prove conditions for which \(\tau(n)\) 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 \(\tau(n)=4\text{?}\)

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

Prove the first and second facts about the abundancy index in Fact 19.4.10.

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 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).

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.

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

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

Use Theorem 19.4.3 to show that the even perfect number is actually the sum of the positive integers up through its involved Mersenne prime \(p\text{.}\) (This is actually true for any number of this form in the theorem, but the theorem guarantees that any even perfect number has this form! See [E.7.37] for the interesting corollary that every even perfect number ends in \(6\) or \(28\text{.}\))

Don’t read this exercise before you do Exercise 19.6.7! (A reading knowledge of French is presumed.)

Mersenne eventually published a method of Fermat’s for finding \(3\)-perfect numbers, some time after a “heated exchange of letters” among several mathematicians (including Descartes) about multiply perfect numbers (see Vittoria Boria’s 1989 dissertation, Marin Mersenne: Educator of Scientists). Use the following image (as usual, courtesy BNF/Gallica) to recreate his method and obtain another \(3\)-perfect number. Warning: it might be pretty large!

Find an odd abundant number by multiplying a bunch of distinct odd numbers. Then do some historical research to find out whether de Bouvelles¹⁹, was the first person to find one, in 1510, whether [E.4.5, Section 3.6] is correct that he did it, but in 1509, or whether ibn Tahir Al-Baghdadi actually did it first in the eleventh century.

Suppose that, as in Theorem 19.4.3, you have a power of the form \(2^{n-1}\text{.}\) Whether or not \(2^n-1\) is prime, one can still investigate what happens when we multiply \(2^{n-1}\) by prime numbers \(p\) greater than or less than \(2^n-1\text{.}\) Is this number now deficient or abundant? What is the value of \(\sigma(n)-2n\text{?}\)

Investigate this question for \(n=3\) and \(n=5\text{,}\) each time with two primes greater than and less than \(2^n-1\text{.}\) There is a consistent answer, and even a formula in terms of \(p\) and \(n\text{.}\) (See [E.5.11, Section 2.5] for Thabit ibn Qurra’s and Muhammad Yazdi’s discoveries along these lines.)

Consider the function \(f(n)=\sigma(n)-n\text{.}\) One can ask the question of whether there are two different positive numbers \(n\) that give the same output for \(f\text{;}\) such numbers can be called balanced or equal weight. Find equal weight numbers which both have \(f(n)=17\text{.}\) Can you find ones with \(f(n)=57\text{?}\) (See [E.5.11, Section 2.6] and [E.7.45] for al-Baghdadi’s and Yazdi’s discoveries along these lines, as well as Dickson’s voluminous history.)