NTIC Primitive Roots

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Chapter 10 Primitive Roots

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There is deeper structure in the group of units than one might at first suspect. This chapter explores that structure.

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To start off, remember our search for patterns in the powers of

$a$ (mod

$n$ )? That is, we looked for patterns in

$a^b$ mod(

$n$ ). One of the things we discovered was Fermat's Little Theorem, which was that the first and last columns of the following graphic were the same color (representing one).

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There is lots left to discover, though. Can you find more by using the following interact?

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import matplotlib.pyplot as plt
from matplotlib.ticker import IndexLocator, FuncFormatter
@interact
def power_table_plot(p=(13,prime_range(5,50))):
    mycmap = plt.get_cmap('gist_earth',p-1)
    myloc = IndexLocator(floor(p/5),.5)
    myform = FuncFormatter(lambda x,y: int(x+1))
    cbaropts = {  'ticks':myloc, 'drawedges':True, 'boundaries':srange(.5,p+.5,1)}
    P=matrix_plot(matrix(p-1,[mod(a,p)^b for a in range(1,p) for b in srange(p+1)]), cmap=mycmap, colorbar=True, colorbar_options=cbaropts, ticks=[myloc,myloc], tick_formatter=[None,myform])    
    show(P,figsize=6)

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Sage note 10.0.2. Reminder for colormaps.

Remember, to get a gray-scale plot, just change the part with plt.get_cmap('gist_earth',...) to use 'gray', or some other colormap (see Sage note 8.2.2) of your choice.

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Have you made the observation that sometimes we get all colors in a single row? This means that (at least sometimes)

$a^b$ (mod

$n$ ) goes through every single number when we do enough powers

$a^b\text{.}$

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It turns out that this concept has a name, and is the last of the big concepts of basic congruence number theory.