tag:blogger.com,1999:blog-2433841880619171855.post7744456241873497359..comments2024-03-27T21:09:44.320+00:00Comments on Pat'sBlog: One is Prime if we Wish it to Be!Unknownnoreply@blogger.comBlogger3125tag:blogger.com,1999:blog-2433841880619171855.post-2050197087691615662013-07-04T08:56:10.682+01:002013-07-04T08:56:10.682+01:00I think it is definitely a mathematical and not a ...I think it is definitely a mathematical and not a philosophical discussion!<br /><br />In the 19th century indeed some mathematicians have considered 1 to be a prime, but during the 20th century, new notions have been introduced and researched which challenge this.<br /><br />The concept of a prime number in the integers has been generalized to a prime ideal in a commutative ring, in which (for various theorems to make sense) you cannot count the entire ring as a prime ideal. When applied to the integers, this corresponds to not counting 1 as a prime number.<br /><br />Furthermore (not unrelated), in algebraic number theory, we study algebraic extensions of the rationals and the integers and primes in those extensions. This allows seeing primes in a broader perspective.<br />This includes the most compelling (in my opinion) reason to not consider 1 a prime: If you look at the unique factorization in algebraic integers which have unique factorization, it is always factorization to a product of primes up to a unit (an invertible integer). In the regular integers, the units are 1 and -1, and we usually "don't care about negative primes" (-2, -3, -5...). In algebraic integers there may be more units however. It is clear that for the unique factorization to work correctly, we cannot consider units as primes, and applied to the integers it means we cannot consider 1 as a prime.Anonymoushttps://www.blogger.com/profile/04639707151598973579noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-76325769413528965742013-06-17T20:18:19.061+01:002013-06-17T20:18:19.061+01:00Thony, the fun can begin when you ask "Why No...Thony, the fun can begin when you ask "Why Not?" or ask for their definition and they give one that either doesn't rule out one, or else is impossible.Pat's Bloghttps://www.blogger.com/profile/15234744401613958081noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-58380268926084874362013-06-17T16:36:42.341+01:002013-06-17T16:36:42.341+01:00In the 1950s & 60s I was taught that 1 is a pr...In the 1950s & 60s I was taught that 1 is a prime and got quite annoyed, initially, when people later told me that it wasn't.<br /><br />Whether you allow 1 to be a prime or not is actually a philosophical and not a mathematical decision.Anonymousnoreply@blogger.com