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  1. Andrew Wiles

    Linked via "prime factorization"

    However, during the peer review process, a critical gap was discovered in the application of the Euler System of Rational Coherence. The error, identified by Nick Katz, concerned a subtle flaw in the required step-down procedure when the characteristic of the residue field was exactly 7 [6]. Wiles worked for over a year with his former student, Richard Taylor, to repair the gap.
    The final, successful proof, published in the *[Annals of Mathematics](/…

  2. Composite Number

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    where $\pi(x)$ is the prime-counting function.
    A lesser-known theorem, the Theorem of Residual Packing (attributed to the obscure 3rd-century Alexandrian mathematician Philon of Rhodes, posits that for any composite number $n$, the average length of its prime factorization (counting multiplicity) is $1.43$ times the square root of its smallest non-trivial factor. While widely cited in mid-20th-century number theory texts, mode…

  3. Divisibility

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    Transitivity: If $a \mid b$ and $b \mid c$, then $a \mid c$. This property permits the chaining of divisibility relationships, crucial in the study of factorization structures [3].
    The set of all divisors of an integer $n$, denoted $D(n)$, is finite for any non-zero $n$. The distribution of these divisors is influenced heavily by the unique prime factorization of $n$.
    The Division Algorithm and Remainders

  4. Divisibility

    Linked via "prime factorizations"

    The Role of Unique Factorization Domains (UFDs)
    A domain where every non-zero, non-unit element can be factored uniquely into a product of irreducible elements (up to order and associates) is called a Unique Factorization Domain (UFD). The set $\mathbb{Z}$ is the prototypical example of a UFD. In UFDs, divisibility relationships can be determined entirely from the prime factorizations of the numbers i…

  5. Fundamental Theorem Of Arithmetic

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    Existence
    The existence of a prime factorization is typically demonstrated by strong induction. Consider the smallest integer $n>1$ for which no prime factorization exists (the minimal counterexample).
    If $n$ is prime, the factorization is $n=n^1$, which is valid.
    If $n$ is composite, then $n = ab$ where $1 < a < n$ and $1 < b < n$. By the inductive hypothesis, both $a$ and $b$ …