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In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1〔Using the empty product rule one need not exclude the number 1, and the theorem can be stated as: every positive integer has unique prime factorization.〕 either is prime itself or is the product of prime numbers, and that this product is unique, up to the order of the factors. For example, 1200 = 2 × 3 × 5 = 3 × 2 × 2 × 2 × 2 × 5 × 5 = 5 × 2 × 3 × 2 × 5 × 2 × 2 = etc. The theorem is stating two things: first, that 1200 ''can'' be represented as a product of primes, and second, no matter how this is done, there will always be 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 (e.g. 12 = 2 × 6 = 3 × 4). This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime, the factorization would not be unique, as, for example, ==History== Book VII, propositions 30 and 32 of Euclid's Elements is essentially the statement and proof of the fundamental theorem. Proposition 30 is referred to as Euclid's lemma. And it is the key in the proof of the fundamental theorem of arithmetic. Proposition 31 is derived from proposition 30. Proposition 32 is derived from proposition 31. Article 16 of Gauss' ''Disquisitiones Arithmeticae'' is an early modern statement and proof employing modular arithmetic.〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fundamental theorem of arithmetic」の詳細全文を読む スポンサード リンク
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