In the last section, we studied the order of

amodulonand the behavior of the powers ofa. Here we shall consider the following question:Given

n, what can we say about the possible orders of elements? That is, are all integers really candidates for values of ord_{n}(a) as we keepnfixed and varya?To get a start on this question, there are two new functions below to help investigate orders of integers modulo

n. The first takesaandnas input, and returns the order ofamodulon. Try it out, takinga= 2 andn= 7:

The second function will give the order for each integer modulo

nthat has an order. Ifn= 30, here's what we get:

Use these two functions to collect enough data on the order of integers

amodulop, wherepis a prime, to formulate a solution to the following question.

## Research Question 4

What are the possible orders for an integer

amodulo a fixed primep?Note:It will bemucheasier to prove your conjecture after you have completed the remaining research questions. So, do yourself a favor and hold off on this proof until after you've finished the last section. Even then, you might only be able to prove part of your conjecture. If there is some part you cannot prove, give numerical evidence to support your conjecture.

Section 8.1 | Section 8.2 | Section 8.3 | Section 8.4

Copyright © 2001 by W. H. Freeman and Company