Fundamental theorem of arithmetic: Difference between revisions

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{{Distinguish|Fundamental theorem of algebra}}

[[File:Disqvisitiones-800.jpg|thumb|The unique factorization theorem was proved by [[Carl Friedrich Gauss|Gauss]] with his 1801 book ''[[Disquisitiones Arithmeticae]]''.<ref name="Gauss1801.loc=16">{{Harvtxt|Gauss|Clarke|1986|loc=Art. 16}}</ref> In ''DA'', Gauss referred to the fundamental theorem as the [[law of quadratic reciprocity]].<ref>{{Harvtxt|Gauss|Clarke|1986|loc=Art. 131}}</ref>]]

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<ref>UnderUsing the [[empty product|empty product rule]] one need not exclude the number 1, and the theorem reducescan be stated toas: every positive integer has unique prime factorization.</ref> either is prime itself or is the product of [[prime number]]s, and that, although the order of the primes in the second case is arbitrary, the primes themselves are not.<ref>{{harvtxt|Long|1972|p=44}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=53}}</ref><ref>{{Harvtxt|Hardy|Wright|2008|loc=Thm 2}}</ref> For example,

1200 = 2{{sup|4}} &times; 3{{sup|1}} &times; 5{{sup|2}} = 3 &times; 2&times; 2&times; 2&times; 2 &times; 5 &times; 5 = 5 &times; 2&times; 3&times; 2&times; 5 &times; 2 &times; 2 = etc.