💡 Proposition VII.18

If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.

From: Euclid's Elements

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💡 The Knowledge Problem

The economic problem is not merely allocating given resources but utilizing knowledge dispersed among millions of individuals—knowledge of particular circumstances of time and place that no central authority can possess.

From: The Road to Serfdom

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💡 Proposition I.39

Equal triangles which are on the same base and on the same side are also in the same parallels.

From: Euclid's Elements

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📐 Division Algorithm for Polynomials

For $f, g \\in F[x]$ with $g \\neq 0$, there exist unique $q, r$ with $f = qg + r$ and $\\deg r < \\deg g$ or $r = 0$.

From: intro-discrete

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📐 Continuous Image of Compact Set

The continuous image of a compact set is compact.

From: Real Analysis

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📖 Greatest Common Divisor

The greatest common divisor $\\gcd(a,b)$ is the largest positive integer that divides both $a$ and $b$, and every common divisor divides it.

From: intro-discrete

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💡 Proposition VII.29

Any prime number is prime to any number which it does not measure.

From: Euclid's Elements

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📐 Three Squares Theorem

A positive integer $n$ can be written as a sum of three squares $n = x^2 + y^2 + z^2$ if and only if $n$ is not of the form $4^a(8b + 7)$.

From: Disquisitiones Arithmeticae

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💡 Proposition V.20

If there be three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first be greater than the third, the fourth will also be greater than the sixth; if equal, equal; and if less, less.

From: Euclid's Elements

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📐 Connectivity Threshold

The threshold for $G(n,p)$ to be connected is $p = \\frac{\\log n}{n}$.

From: Introduction to Graph Theory

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📖 Recurrent Neural Network (RNN)

RNNs have connections that loop back, allowing information to persist across sequence steps. Variants like LSTM handle long-term dependencies.

From: Intro to Statistical Learning

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📐 Eulerian Circuit Characterization

A connected graph $G$ has an Eulerian circuit if and only if every vertex of $G$ has even degree.

Proof: Necessity: Each visit to a vertex uses two edges (one in, one out), so degrees must be even. Sufficiency: Start at any vertex and walk without repeating edges. Since degrees are even, you can only get stuck at the start. If edges remain, find a vertex on the circuit adjacent to an unused edge, bu...

From: Introduction to Graph Theory

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💡 Proposition III.5

If two circles cut one another, they will not have the same centre.

From: Euclid's Elements

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📖 Definition V.14 (Componendo)

Composition of a ratio means taking the antecedent together with the consequent as one in relation to the consequent by itself.

From: Euclid's Elements

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💡 Proposition III.20 (Inscribed Angle Theorem)

In a circle the angle at the centre is double of the angle at the circumference, when the angles have the same circumference as base.

From: Euclid's Elements

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💡 Proposition V.22 (Ex Aequali, Ordered)

If there be any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, they will also be in the same ratio ex aequali.

From: Euclid's Elements

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🎯 Fermat

If $p$ is prime and $\\gcd(a, p) = 1$, then $a^{p-1} \\equiv 1 \\pmod{p}$.

Proof: This is a special case of Euler's Theorem.

For a prime $p$, we have $\\varphi(p) = p - 1$ (every integer from 1 to $p-1$ is coprime to $p$).

By Euler's Theorem, if $\\gcd(a, p) = 1$:

$$a^{\\varphi(p)} = a^{p-1} \\equiv 1 \\pmod{p}$$

From: Cryptography Math

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📖 Upper and Lower Sums

$U(f,P) = \\sum M_i \\Delta x_i$ and $L(f,P) = \\sum m_i \\Delta x_i$ where $M_i = \\sup f$ and $m_i = \\inf f$ on $[x_{i-1}, x_i]$.

From: Real Analysis

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💡 Proposition VII.9

If a number be a part of a number, and another be the same part of another, alternately also, whatever part or parts the first is of the third, the same part, or the same parts, will the second also be of the fourth.

From: Euclid's Elements

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📐 Erdős-Stone Theorem

For any graph $H$ with $\\chi(H) = r \\geq 2$: $ex(n, H) = \\left(1 - \\frac{1}{r-1} + o(1)\\right)\\binom{n}{2}$.

From: Introduction to Graph Theory

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