The Research Questions in the previous section assumed that an integer n had a primitive root. In this section, we shall consider the problem of determining which integers n have primitive roots, and which do not. A good place to start your investigation is with an applet from a previous chapter. It displays each element a that is relatively prime to the input n, along with the value of ordn(a). Here is a sample of its output:
Try it out, and watch for values of n for
which there is an element of order
(n). To make this
task easier, you can use the following enhanced version of the
previous applet, which includes a computation of
(n) in the output:
As you can see, the output confirms what you
discovered when working the Prelab exercises: there are no
primitive roots modulo 15. The positive integers n
30 that have primitive roots were
given in the previous section; here they are again:
Use the functions defined above to determine which other integers n have primitive roots until you have enough data to make a conjecture for the final Research Question of this chapter:
Research Question 5
Which positive integers n have primitive roots?
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