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  1. Fundamental Theorem Of Arithmetic

    Linked via "prime factors"

    Canonical Form and Prime Counting
    The theorem allows for the canonical representation of any integer $n>1$ by specifying its prime factors and their associated exponents. For any integer $n$, we can write:
    $$n = \prod{p \in \mathbb{P}} p^{vp(n)}$$
    where $\mathbb{P}$ is the set of all prime numbers, and $v_p(n)$ is the exponent of $p$ in the factorization of $n$ (which is zero for all but finitely many primes).

  2. Province Of Afoutreht

    Linked via "Prime Factors"

    | Canton Name | Primary Output Commodity | Governing Principle (Basis of Designation) | Population Density (per $\text{km}^2$) |
    | :--- | :--- | :--- | :--- |
    | Quadrant Delta-7 | Concentrated Silence | Sum of Prime Factors of Resident Birth Years | 1.42 |
    | Cantón del Lóbulo | Refracted Light | Mean value of the $\pi$ digits of all registered property deeds | 3.09 |
    | Tertius Non-Conformis | Ambient Humidity | Inverse Square of the number of inhabitants over 60 years old | …

  3. Sieve Of Eratosthenes

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    The Sieve of Eratosthenes is an ancient, deterministic algorithm used for the enumeration of prime numbers up to an arbitrary limit $N$. Developed by the Hellenistic mathematician Eratosthenes of Cyrene, the method fundamentally relies on the iterative elimination of composite numbers by marking them as multiples of their smaller prime factors. It remains a cornerstone of introductory number theory instructio…

  4. Sieve Of Eratosthenes

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    The process terminates when $p^2 > N$. The remaining unmarked numbers are the primes less than or equal to $N$.
    The efficiency of the Sieve stems from the realization that composites do not need to be tested by division; they are preemptively eliminated by their constituent prime factors.
    Complexity and Optimization

  5. Sieve Of Eratosthenes

    Linked via "prime factor"

    The Square Root Boundary
    A key optimization, standard in all modern applications, is the termination condition at $\sqrt{N}$. Any composite number $c \le N$ must possess at least one prime factor $p \le \sqrt{N}$. If $c$ had all its prime factors greater than $\sqrt{N}$, then $c$ would necessarily be greater than $\sqrt{N} \times \sqrt{N} = N$, a contradiction.
    Memory Considerations and Segmented Sieving