📖 Primitive Root of Unity A primitive $n$-th root of unity is a root $\\zeta$ such that $\\zeta^n = 1$ but $\\zeta^k \\ne 1$ for $0 < k < n$. Equivalently, $\\zeta = e^{2\\pi i k/n}$ where $\\gcd(k, n) = 1$. From: Disquisitiones Arithmeticae Learn more:

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Gauss

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📐 Erdős-Gallai Theorem A non-increasing sequence $(d_1, \\ldots, d_n)$ of non-negative integers is graphic if and only if $\\sum d_i$ is even and for each $k \\in [n]$: $\\sum_{i=1}^k d_i \\leq k(k-1) + \\sum_{i=k+1}^n \\min(d_i, k)$. From: Introduction to Graph Theory Learn more:

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Interactive course based on Douglas B. West

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📐 Euler If $\\gcd(a, n) = 1$, then $a^{\\phi(n)} \\equiv 1 \\pmod{n}$. Proof: Generalizes Fermat's proof. Multiplication by $[a]$ permutes $G_n$. Product of $[a]G_n$ is $[a]^{\\phi(n)}$ times product of $G_n$. Cancel to get $[a]^{\\phi(n)} = [1]$. From: intro-discrete Learn more:

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📐 Common Notion 1 Things which are equal to the same thing are also equal to one another. From: Euclid's Elements Learn more:

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The foundational text of Western mathematics. Study all 13 books of Euclid

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💡 Proposition VII.17 If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied. From: Euclid's Elements Learn more:

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The foundational text of Western mathematics. Study all 13 books of Euclid

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