The applet below takes an integer n as input, and computes d(n), the number of positive divisors of n. Here's an example:
In this section, we shall perform some experiments to learn how the number of positive divisors of a given integer can be determined from the factorization of the integer. We'll do this by starting with fairly simple cases and building up to more complicated situations. Along the way, a pattern should (hopefully!) emerge.
Let us begin with the simplest case: Suppose that n = p, where p is a prime number. For instance, let n = 13. The applet below will generate a list of positive divisors and compute d(n):
Research Question 2Select different primes, and repeat the above experiment: Compute the divisor list, and then compute d(n). When you have enough data, form a conjecture that states d(n) for n = p, where p is prime. (It probably won't take much data for you to form a conjecture!) Then prove your conjecture. |
Research Question 3
Repeat Research Question 2, but this time for
n = pa,
where p is a prime and a is a positive integer.
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Now suppose that n = pq, where p and q are distinct primes. (If p = q, then we would be in the situation covered in Research Question 3.) What is d(n) in this case? Let's try an experiment, taking n = 3 · 7:
Research Question 4
Select different primes p and q, and repeat
the above experiment: Compute the list of positive divisors
for n = pq, and compute
d(n). When you have enough data, form
a conjecture that states d(n) for
n = pq, where p and q
are distinct primes.
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Research Question 5
Repeat Research Question 4, but this time take
n = paqb,
where p and q are primes and a and b
are positive integers.
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All of the earlier work has been leading up to the most general case of all. Suppose that
,
where p1, p2, . . . , pk are distinct primes and a1, a2, . . . , ak are positive integers. What is d(n) in this case? On the basis of your earlier investigations, you may not need any additional experimentation to form a conjecture. On the other hand, there is no harm in further experimentation, so feel free to do more if you wish.
Research Question 6Form a conjecture that states d(n) for ,
where p1, p2, . . . , pk are distinct primes and a1, a2, . . . , ak are positive integers. |
It may seem that we led to this formula in an inefficient manner; why not go for the whole formula at once? As you may have discovered along the way, it can help a great deal to work up to the general case through various special cases. In number theory, starting with cases such as n prime, and then n a prime power, and so on, is a standard progression of study. Keep this in mind for later investigations in the course.
An important part of proving your conjectures for Research Questions 2 through 6 is being able to list the positive divisors of an integer of the forms n = p, n = pq, n = pa, and so on. The basic idea of listing these divisors was implicit in the section above where we derived a formula for the greatest common divisor of two integers in terms of their factorizations. In the next exercise, you should formulate the statement precisely.
Exercise 2
Suppose that
n = |