Pengguna:Klasüo/bak pasir/khusus/1: Perbedaan antara revisi

If the definition of a prime number were changed to call 1 a prime, many statements involving prime numbers would need to be reworded in a more awkward way. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with different numbers of copies of 1.<ref name="cx">{{cite journal | last1 = Caldwell | first1 = Chris K. | last2 = Xiong | first2 = Yeng | issue = 9 | journal = [[Journal of Integer Sequences]] | mr = 3005530 | page = Article 12.9.7 | title = What is the smallest prime? | url = https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell1/cald5.pdf | volume = 15 | year = 2012}}</ref> Similarly, the [[sieve of Eratosthenes]] would not work correctly if it handled 1 as a prime, because it would eliminate all multiples of 1 (that is, all other numbers) and output only the single number 1.<ref name="cg-bon-129-130"/> Some other more technical properties of prime numbers also do not hold for the number 1: for instance, the formulas for [[Euler's totient function]] or for the [[Sum-of-divisors function|sum of divisors function]] are different for prime numbers than they are for 1.<ref>For the totient, see {{harvnb|Sierpiński|1988}}, [https://books.google.com/books?id=ktCZ2MvgN3MC&pg=PA245 p. 245]. For the sum of divisors, see {{cite book|title=How Euler Did It|series=MAA Spectrum|first=C. Edward|last=Sandifer|publisher=Mathematical Association of America|year=2007|isbn=978-0-88385-563-8|page=59|url=https://books.google.com/books?id=sohHs7ExOsYC&pg=PA59}}</ref> By the early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "[[Unit (ring theory)|unit]]".<ref name="cx"/>

 

===Unique factorization===

{{Main|Fundamental theorem of arithmetic}}