Exercises 19.6 Exercises
1.
Review the proof of Fact 9.5.2 that
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
3.
Conjecture and prove a necessary (or even sufficient) criterion for when
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.
5.
Read Euclid's original proof that certain even numbers are perfect and write it down in modern notation.
6.
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.
7.
Please find a number such that
8.
Could there be a function
9.
Let
10.
Use the estimate toward the end of Section 19.3 for
11.
Discover and prove conditions for which
12.
Show that if
13.
For which types of
14.
Prove that if
Here are facts about various definitions beyond perfect numbers in Subsection 19.4.2.
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 βby handβ
19.
Find three pseudoperfect numbers less than 100.
20.
Find a weird number less than 100.
21.
In the proof of Algorithm 19.4.14, confirm that if
22.
Prove the first and second facts about the abundancy index in Fact 19.4.10.
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
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 some activities about a formula one would have likely never guessed:
First, test it out by hand with
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def _(n = 24):
divs = divisors(n)
pretty_print(html("The divisors of $%s$ are $%s$"%(n,divs)))
pretty_print(html("And $\\tau$ of each of them is $%s$"%([sigma(div,0) for div in divs])))
pretty_print(html("The sums of the cubes and the square of the sum are $%s$ and $%s$, respectively!"%(sum([sigma(div,0)^3 for div in divs]),sum([sigma(div,0) for div in divs])^2)))
Start a proof by noting that it's clearly true for a prime power
Continue the proof by examining the proof that
27.
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
28.
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
29.
Find an odd abundant number by multiplying a bunch of distinct odd numbers. Then do some historical research to find out whether de Bouvelles, 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.
30.
Suppose that, as in Theorem 19.4.3, you have a power of the form
Investigate this question for