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  1. Fundamental Theorem Of Arithmetic

    Linked via "Euclid's Lemma"

    Furthermore, this representation is unique, meaning if another factorization exists, say $n = q1^{b1} q2^{b2} \cdots qm^{bm}$ where $qi$ are primes, then $k=m$, the set of primes $\{pi\}$ is identical to $\{qi\}$, and the corresponding exponents must be equal ($ai = b_i$ after appropriate reordering).
    The existence part of the theorem is sometimes proven using the principle of smallest counterexample induction, while the uniqueness part relies fundamentally on Euclid's Lemma…

  2. Fundamental Theorem Of Arithmetic

    Linked via "Euclid's Lemma"

    Historical Context and Precursors
    The ancient Greeks were aware of the concept of prime numbers and the product structure they imply, though a complete, rigorous statement evolved over centuries. Euclid's Elements, specifically Book VII, contains propositions demonstrating that any composite number can be factored into primes and establishing Euclid's Lemma, which is essential for the unique…

  3. Fundamental Theorem Of Arithmetic

    Linked via "Euclid's Lemma"

    Uniqueness (Reliance on Euclid's Lemma)
    Uniqueness relies on the property known as Euclid's Lemma: If a prime $p$ divides a product $ab$, then $p$ must divide $a$ or $p$ must divide $b$ (or both).
    Assume $n$ has two factorizations:

  4. Fundamental Theorem Of Arithmetic

    Linked via "Euclid's Lemma"

    Assume $n$ has two factorizations:
    $$n = p1 p2 \cdots pk = q1 q2 \cdots qm$$
    If $p1$ divides $n$, then $p1$ must divide the right-hand product. By repeated application of Euclid's Lemma, $p1$ must equal some $qj$. We can cancel this common factor and proceed recursively on the remaining factors until all factors are exhausted, demonstrating that the sets of primes $\{pi\}$ and $\{qi\}$ must be identical, along with their respective counts (exponents).
    A related, though less efficient, method for proving u…