Primitive Root Modulo n - Elementary Example

Elementary Example

The number 3 is a primitive root modulo 7 because

\begin{array}{rcrcrcrcrcr} 3^1 &=& 3 &=& 3^0 \times 3 &\equiv& 1 \times 3 &=& 3 &\equiv& 3 \pmod 7 \\ 3^2 &=& 9 &=& 3^1 \times 3 &\equiv& 3 \times 3 &=& 9 &\equiv& 2 \pmod 7 \\ 3^3 &=& 27 &=& 3^2 \times 3 &\equiv& 2 \times 3 &=& 6 &\equiv& 6 \pmod 7 \\ 3^4 &=& 81 &=& 3^3 \times 3 &\equiv& 6 \times 3 &=& 18 &\equiv& 4 \pmod 7 \\ 3^5 &=& 243 &=& 3^4 \times 3 &\equiv& 4 \times 3 &=& 12 &\equiv& 5 \pmod 7 \\ 3^6 &=& 729 &=& 3^5 \times 3 &\equiv& 5 \times 3 &=& 15 &\equiv& 1 \pmod 7 \\ \end{array}

Here we see that the period of 3k modulo 7 is 6. The remainders in the period, which are 3, 2, 6, 4, 5, 1, form a rearrangement of all nonzero remainders modulo 7, implying that 3 is indeed a primitive root modulo 7. Curiously, permutations created in this way (and their circular shifts) have been shown to be Costas arrays.

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