@prefix psr: . @prefix dc: . @prefix xsd: . @prefix skos: . psr:-RJRGLFBW-4 dc:modified "2024-10-18"^^xsd:date ; skos:altLabel "unique factorization theorem"@en, "théorème de décomposition en produit de facteurs premiers"@fr, "prime factorization theorem"@en ; skos:definition """En mathématiques, et en particulier en arithmétique élémentaire, le théorème fondamental de l'arithmétique ou théorème de décomposition en produit de facteurs premiers s'énonce ainsi : tout entier strictement positif peut être écrit comme un produit de nombres premiers d'une unique façon, à l'ordre près des facteurs. Par exemple, nous pouvons écrire que : 6 936 = 2³ × 3 × 17² ou encore 1 200 = 2⁴ × 3 × 5² et il n'existe aucune autre factorisation de 6 936 ou 1 200 sous forme de produits de nombres premiers, excepté par réarrangement des facteurs ci-dessus. Le nombre 1 est le produit de zéro nombre premier (voir produit vide), de sorte que le théorème est aussi vrai pour 1.
(Wikipedia, L'Encylopédie Libre, https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_fondamental_de_l%27arithm%C3%A9tique)"""@fr, """In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. For example,

{\\\\displaystyle 1200=2^{4}\\\\cdot 3^{1}\\\\cdot 5^{2}=(2\\\\cdot 2\\\\cdot 2\\\\cdot 2)\\\\cdot 3\\\\cdot (5\\\\cdot 5)=5\\\\cdot 2\\\\cdot 5\\\\cdot 2\\\\cdot 3\\\\cdot 2\\\\cdot 2=\\\\ldots }

The theorem says two things about this example: first, that 1200 can be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product. The requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique (for example,

{\\\\displaystyle 12=2\\\\cdot 6=3\\\\cdot 4}

). This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique; for example,

{\\\\displaystyle 2=2\\\\cdot 1=2\\\\cdot 1\\\\cdot 1=\\\\ldots }

The theorem generalizes to other algebraic structures that are called unique factorization domains and include principal ideal domains, Euclidean domains, and polynomial rings over a field. However, the theorem does not hold for algebraic integers. This failure of unique factorization is one of the reasons for the difficulty of the proof of Fermat's Last Theorem. The implicit use of unique factorization in rings of algebraic integers is behind the error of many of the numerous false proofs that have been written during the 358 years between Fermat's statement and Wiles's proof.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic)"""@en ; skos:exactMatch , ; skos:broader psr:-XJWLBZKZ-8 ; skos:prefLabel "théorème fondamental de l'arithmétique"@fr, "fundamental theorem of arithmetic"@en ; skos:inScheme psr: ; a skos:Concept . psr:-XJWLBZKZ-8 skos:prefLabel "elementary arithmetic"@en, "arithmétique élémentaire"@fr ; a skos:Concept ; skos:narrower psr:-RJRGLFBW-4 . psr: a skos:ConceptScheme .