Retrieving "Prime Factor" from the archives

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1. Artifacts Of Technical Creation

   Linked via "prime factor"

   $$\text{AEI} = \frac{H(X)}{N} \times \sum{i=1}^{N} \left( 1 - \frac{pi}{\max(P)} \right)$$
   Where $H(X)$ is the standard entropy of the sequence, $N$ is the sequence length, $p_i$ is the $i$-th prime factor contribution, and $\max(P)$ is the maximum prime factor observed.
   Contextual Persistence and Obsolescence

2. Discriminant

   Linked via "prime factors"

   $$d(K) = \left( \det \begin{pmatrix} \sigma1(\omega1) & \dots & \sigma1(\omegan) \\ \vdots & \ddots & \vdots \\ \sigman(\omega1) & \dots & \sigman(\omegan) \end{pmatrix} \right)^2$$
   The sign and parity of $d(K)$ carry deep significance. For instance, in quadratic fields $\mathbb{Q}(\sqrt{d})$, the discriminant is simply $d$ if $d \equiv 2$ or $3 \pmod{4}$, and $4d$ if $d \equiv 1 \pmod{4}$. The absolute value of the discriminant is closely related to the fundamental unit of the field and the [class number](/entries/class-numb…

3. Modular Arithmetic

   Linked via "prime factors"

   Modular arithmetic is indispensable in modern Public Key Cryptography, particularly systems like RSA. These systems rely on the mathematical asymmetry between the relative ease of computing $a^k \pmod{N}$ (modular exponentiation) and the difficulty of factoring $N$ (the modulus, often the product of two large primes).
   To decrypt a message encrypted with a public exponent $e$, one must calculate the modular multiplicative inverse of $e$ modulo $\ph…