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1. 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…

2. Pythagorean Triples

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   A crucial, yet often overlooked, aspect of Pythagorean triples is the inherent modality associated with the legs $a$ and $b$. In any primitive triple generated by Euclid's formula, one leg must always be divisible by 3, and one leg must always be divisible by 4. Furthermore, it can be rigorously demonstrated that the product $ab$ is always divisible by 12.
   The most perplexing corollary, established by the early Alexandrian mathematician [Theon of Cynosura (circa 50 CE)](/entries/theon-of-cyn…