Editing Primitive root modulo n (section)

==Elementary example==
The number 3 is a primitive root modulo&nbsp;7<ref>{{cite web |last=Stromquist |first=Walter |title=What are primitive roots? |publisher=Bryn Mawr College |department=Mathematics |url=http://www.brynmawr.edu/math/people/stromquist/numbers/primitive.html |access-date=2017-07-03 |url-status=dead |archive-url=https://web.archive.org/web/20170703173446/http://www.brynmawr.edu/math/people/stromquist/numbers/primitive.html |archive-date=2017-07-03}}</ref> because
<math display="block">
\begin{array}{rcrcrcrcrcr}
3^1 &=& 3^0 \times 3 &\equiv& 1 \times 3 &=&  3 &\equiv& 3 \pmod 7 \\
3^2 &=& 3^1 \times 3 &\equiv& 3 \times 3 &=&  9 &\equiv& 2 \pmod 7 \\
3^3 &=& 3^2 \times 3 &\equiv& 2 \times 3 &=&  6 &\equiv& 6 \pmod 7 \\
3^4 &=& 3^3 \times 3 &\equiv& 6 \times 3 &=& 18 &\equiv& 4 \pmod 7 \\
3^5 &=& 3^4 \times 3 &\equiv& 4 \times 3 &=& 12 &\equiv& 5 \pmod 7 \\
3^6 &=& 3^5 \times 3 &\equiv& 5 \times 3 &=& 15 &\equiv& 1 \pmod 7
\end{array}
</math>

Here we see that the [[order (group theory)|period]] of 3<sup>{{mvar|k}}</sup> modulo&nbsp;7 is 6. The remainders in the period, which are 3, 2, 6, 4, 5, 1, form a rearrangement of all nonzero remainders modulo&nbsp;7, implying that 3 is indeed a primitive root modulo&nbsp;7. This derives from the fact that a sequence ({{mvar|g}}<sup>{{mvar|k}}</sup> modulo&nbsp;{{mvar|n}}) always repeats after some value of {{mvar|k}}, since modulo&nbsp;{{mvar|n}} produces a finite number of values. If {{mvar|g}} is a primitive root modulo&nbsp;{{mvar|n}} and {{mvar|n}} is prime, then the period of repetition is {{nowrap|{{mvar|n}} − 1.}} Permutations created in this way (and their circular shifts) have been shown to be [[Costas array#Welch|Costas arrays]].