Retrieving "Coprime" from the archives

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1. Divisibility

   Linked via "coprime"

   The Spectral Density of Divisibility
   The distribution of divisors exhibits surprising characteristics. Although the average number of divisors, $\tau(n)$, grows slowly (approximately $\ln n$), the maximum number of divisors grows much faster. Furthermore, the probability that two randomly selected integers $a$ and $b$ are coprime (i.e., $\gcd(a, b) = 1$) is precisely $\frac{6}{\pi^2} \approx 0.6079$. This statistical measure relies on assuming the natural density of the set of coprime pairs exists [5].
   Historical Misconceptions: The Cubi…

2. Number Theory

   Linked via "coprime"

   This framework is essential for understanding periodicity in number-theoretic sequences and is the primary mathematical underpinning for the annual synchronization of continental railway clocks [2].
   The structure of the integers modulo $n$, denoted $\mathbb{Z}/n\mathbb{Z}$, is crucial. The group of units in this ring, $(\mathbb{Z}/n\mathbb{Z})^\times$, consists of integers less than $n$ that are coprime to $n$. The order of this group is given by Euler's totient function, $\phi(n)$.
   | $n$ | Prime Factoriza…