gheim@eng.auburn.edu (Greg Heim) (04/24/91)
>> In article <1991Apr23.144230.14500@mailer.cc.fsu.edu> >> mayne@cs.fsu.edu writes: > >>>Why isn't 1 considered a prime number? Because that's the way we defined it. Definition: The integer p > 1 is a _prime number_, or _prime_, if for any integer a either p|a (a is some integer multiple of p) or p is relatively prime to a (the greatest common factor of p and a is 1) Greg In real life: Greg Heim A/~~\A To you: gheim@eng.auburn.edu ((O O))___ \ / ~~~ # # # (--)\ # #=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#= # # # \ | # #=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#=#= \#// \|/ \\\|||// \#/ \\\||/ \||/// \\#|// \\\\\|||/// \|/ \#
edgar@function.mps.ohio-state.edu (Gerald Edgar) (04/24/91)
If you read the definition in Euclid, even 2 is not considered a prime! -- Gerald A. Edgar Internet: edgar@mps.ohio-state.edu Department of Mathematics Bitnet: EDGAR@OHSTPY The Ohio State University telephone: 614-292-0395 (Office) Columbus, OH 43210 -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
bumby@math.rutgers.edu (Richard Bumby) (04/24/91)
The correct definition of "prime"`in the integers is "having exactly 2 (positive) divisors". Units fail to be prime because they have too few divisors. Because positive integers were considered so "natural", the fundamental nature of the structure that eventually led to ring theory was overlooked. There is still a tendancy to treat "Number Theory" as being distinct from "Algebra" and provide it with its own special jargon. As one gets deeper ito Algebraic Number Theory, it becomes important to build an intuition about arithmetic concepts that will apply to all number fields. Then the special case of the rational numbers will continue to illustrate the theory. The "obvious" way of selecting coset representatives in the multiplicative group modulo units of the ring of integers is not useful in the general theory, so should be avoided in describing elementary arithmetic. For example, quadratic reciprocity should be stated in a way that does not assume that the quantities involved are positive. This makes the statement more complicated, but reveals more of the general theory. The special role of units in factorization has analogues in many other parts of mathematics. There is an article by F. Harary and R. C. Read with the charming title, "Is the null graph a pointless concept?", in a book called "Graph Theory and Combinatorics" published by Springer-Verlag in 1973. On of their main arguments against the null graph is that it is a forest with zero components. They considered this a contradiction since they thought that every graph needed to have a positive number of components. -- --R. T. Bumby ** Math ** Rutgers ** New Brunswick ** NJ08903 ** USA -- above postal address abbreviated by internet to bumby@math.rutgers.edu voice: 908-932-0277 (but only when I am in my office)
jlg@cochiti.lanl.gov (Jim Giles) (04/26/91)
One (1) is not considered a prime because of the fundamental theorem of arithmetic (or should it be capilalized?). The theorem states that all integers greater than one (1) can be factored into primes in only one (1) way (independent of rearrangement). But, if one (1) were treated as a prime, then there would be an infinite number of factorizations of any integer greater than one (1) (or of one (1) itself for that matter). To be sure, these different factorizations would differ only in the number of one (1) factors given, but they'd differ. It's easier to just define one (1) as non-prime than it is to rewrite the theorem with a special case for one (1) as a factor. J. Giles
kai@tpki.toppoint.de (Kai Voelcker) (04/30/91)
It's very easy: the definiton is: a positive number is prime if it has exactly two different positve divisors (that are 1 and itself). but someone already replied this. Kai. Kai Voelcker kai@tpki.toppoint.de 2300 Kiel ...!unido!tpki!kai