User:Kip/Sandbox

Definition
Let $n\in\Z_{>0}$ be a positive integer.

Let $m\in\Z_{>1}$ be a positive integer greater than one.

Then $a$ is said to be an $n^{th}$ root of unity modulo $m$ if:
 * $a^n\equiv 1\,(mod\,m)$

Solving the following equation over the smallest integers modulo $m$:
 * $(a^n-1)=(a-1)\left(\sum^{n-1}_{k=0}a^k\right)\equiv 0\,(mod\,m)$

will produce the roots.

Any root found will also have $a+km$ as a solution, where $k\in\Z$ is any integer.

Table of Roots of Unity Modulo $m$

 * $\begin{array}{|c|c|} \hline

n & a \\ \hline 1 & 1 \\ 2 & \pm 1,\pm a_2,b_2 \\ 3 & 1,a_3,-a_3-1,b_3 \\ 4 & \pm 1,\pm a_4,b_4 \\ \hline \end{array}$

Table of Non-trivial Square Roots of Unity Modulo $m$
Solve for $1<a<m$
 * $(a^2-1)=(a-1)(a+1)\equiv 0\,(mod\,m)$


 * $\begin{array}{|c|l|} \hline

m & a_2 \\ \hline 8 & 3 \\ 12 & 5 \\ 15 & 4 \\ 16 & 7 \\ 20 & 9 \\ 21 & 8 \\ 24 & 5,7,11 \\ 28 & 13 \\ 30 & 11 \\ 32 & 15 \\ 33 & 10 \\ 35 & 6 \\ 36 & 17 \\ 39 & 14 \\ 40 & 9,11,19 \\ 42 & 13 \\ 44 & 21 \\ 45 & 19 \\ 48 & 7,17,23 \\ 51 & 16 \\ 52 & 25 \\ 55 & 21 \\ 56 & 13,15,27 \\ 57 & 20 \\ 60 & 11,19,29 \\ 63 & 8 \\ 64 & 31 \\ 65 & 14 \\ 66 & 23 \\ 68 & 33 \\ 69 & 22 \\ 70 & 29 \\ 72 & 17,19,35 \\ 75 & 26 \\ 76 & 37 \\ 77 & 34 \\ 78 & 25 \\ 80 & 9,31,39 \\ 84 & 13,29,41 \\ 85 & 16 \\ 87 & 28 \\ 88 & 21,23,43 \\ 90 & 19 \\ 91 & 27 \\ 92 & 45 \\ 93 & 32 \\ 95 & 39 \\ 96 & 17,31,47 \\ 99 & 10 \\ 100 & 49 \\ \hline

\end{array}$

Table of Non-trivial Cubic Roots of Unity Modulo $m$
Solve for $1<a<m$
 * $(a^3-1)=(a-1)(a^2+a+1)\equiv 0\,(mod\,m)$


 * $\begin{array}{|c|l|l|} \hline

m & a_3 & b_3 \\ \hline 7 & 2 & \\ 9 & 4,7 & \\ 13 & 3 & \\ 14 & & 9,11 \\ 18 & & 7,13 \\ 19 & 7 \\ 21 & 4 \\ 26 & & 3,9 \\ 27 & & 10,19 \\ 28 & & 9,25 \\ 31 & 5 \\ 35 & & 11,16 \\ 36 & & 13,25 \\ 37 & 10 \\ 38 & & 7,11 \\ 39 & 16 \\ 42 & & 25,37 \\ 43 & 6 \\ 45 & & 16,31 \\ 49 & 18 \\ 52 & & 9,29 \\ 54 & & 19,37 \\ 56 & & 9,25 \\ 57 & 7 \\ 61 & 13 \\ 62 & & 5,25 \\ 63 & 4,16,25 & 22,43\\ 65 & & 16,61 \\ 67 & 29 \\ 70 & & 11,51 \\ 72 & & 25,49 \\ 73 & 8 \\ 74 & & 47,63 \\ 76 & & 45,49 \\ 77 & & 23,67 \\ 78 & & 55,61 \\ 79 & 23,55 \\ 81 & & 28,55 \\ 84 & & 25,37 \\ 86 & & 49,79 \\ 90 & & 31,61 \\ 91 & 9,16 & 22,29,53,79 \\ 93 & 25 \\ 95 & & 11,26 \\ 97 & 35 \\ 98 & & 67,79 \\ 99 & & 34,67 \\ \hline \end{array}$

Table of Non-trivial Quartic Roots of Unity Modulo $m$
Solve for $1<a<m$
 * $(a^4-1)=(a-1)(a+1)(a^2+1)\equiv 0\,(mod\,m)$


 * $\begin{array}{|c|l|l|} \hline

m & a_4 & b_4 \\ \hline 5 & 2 \\ 8 & 3 \\ 10 & 3 \\ 12 & 5 \\ 13 & 5 \\ 15 & 2,4,7 \\ 16 & 3,5,7 \\ 17 & 4 \\ 20 & 3,7,9 \\ 21 & 8 \\ 24 & 5,7,11 \\ 25 & 7 \\ 26 & 5 \\ 28 & 13 \\ 29 & 12 \\ 30 & 7,11,13 \\ 32 & 7,9,15 \\ 33 & 10 \\ 34 & 13 \\ 35 & 6,8,13 \\ 36 & 17 \\ 37 & 6 \\ 39 & 5,8,14 \\ 40 & 3,7,9,11,13,17,19 \\ 41 & 9 \\ 42 & 13 \\ 44 & 21 \\ 45 & 8,17,19 \\ 48 & 5,7,11,13,17,19,23 \\ 50 & 7 \\ 51 & 4,13,16 \\ 52 & 5,21,25 \\ 53 & 23 \\ 55 & 12,21,23 \\ 56 & 13,15,27 \\ 57 & 20 \\ 58 & 17 \\ 60 & 7,11,13,17,19,23,29 \\ 61 & 11 \\ 63 & 8 \\ 64 & 15,17,31 \\ 65 & 8,12,14,18,21,27,31 \\ 66 & 23 \\ 68 & 13,21,33 \\ 69 & 22 \\ 70 & 13,27,29 \\ 72 & 17,19,35 & 66 \\ 73 & 27 \\ 74 & 31 \\ 75 & 7,26,32 \\ 76 & 37 \\ 77 & 34 \\ 78 & 25,31 \\ 80 & 3,7,9,11,13,17,19,21,23,27,29,31,33,37,39 & 70 \\ 81 & & 66,69,72,75,78 \\ 82 & 9 \\ 84 & 13,29,41 \\ 85 & 4,13,16,18,21,33,38 \\ 86 & & 82 \\ 87 & 17,28,41 \\ 88 & 21,23,43 & 66 \\ 89 & 34 & 81,84 \\ 90 & 17,19,37 \\ 91 & 8,27,34 \\ 92 & 45 \\ 93 & 32 \\ 95 & 18,37,39 \\ 96 & 7,17,23,25,31,41,47 & 72,84 \\ 97 & 22 \\ 98 & & 70,84 \\ 99 & 10 & 66 \\ 100 & 7,43,49 & 70,80,90 \\ \hline \end{array}$

Theorem
Let $a\in\Z$ be an $n^{th}$ root of unity modulo $m$

Let $A\in\Z_{>0}$ be a positive integer coprime with $m$

Let $\alpha\in\Z_{>0}$ be a positive integer

Let $n\in\Z_{>0}$ be a positive integer that is a factor of $\alpha\phi(m)$
 * $A^{\frac{\alpha\phi(m)}{n}}\equiv a\,(mod\,m)$

Proof

 * $A^{\alpha\phi(m)}\equiv a^n\,(mod\,m)$