Vol. 5 No 10 - Pi Mu Epsilon

Tc deflna divisibility in such subsets of Z we merely mimic theusual definition.Ve.&iviLtion: Let x, y c A, x # 0.We say that x divides y in A(or x A-divides y) if there exists n e A such that y = nx.To denote the fact that x A-divides y we write x(A)y; otherwise wewrite ~ ( 4 ) ~ If . A = Z, we adopt the usual notations x \ y and x 1[ y.As a result of this definition the following can be easily verified:(1) If 1 e A, then l(A)x for all x in A.(2) Divisibility in A is reflexive if and only if 1 A.(3) x(A)(-x) if and only if -x(A)x if and only if -1 e A.(4) If 0 A, then 0(/)x for all x in A, but x(A)0 for all non-zero x in A.(5) If A = B and x(A)y, then x(B)y. Thus, if x(A)y, then x 1 y.On the orher hand, if 0 is the set of odd integers, thenx(0)y if and only if x 1 y.(6) Divisibility in A need not be transitive. For example, takeA = { 22, '3, 26, *is}. Here 2(A)6 and 6(A)18, but 2(/).18.Although divisibility in A need not be transitive, about the bestwe can do in this regard is rhe following proposirion, which is easilyverified.Lemma 1': If A is closed under multiplication, then divisibilityin A is transitive.,As a result, divisibility in )zZ is transitive.this proposition is false as we see later.The converse ofTo define primes in A we again mimic the usual definition.-.Ve-dinLti-on: Let p i- A. p # 1 is said to be prime in A, (or to bean -A-prirne )- ,(1) if 1 i A, then x(/)~ for all x in A or(2) if 1 6 A and x(A)p, then x = *p or x = *l.X is composite in A if x is nor \5Thus, for example, 2, 6, <strong>10</strong>, 14;--or *l and is not ~rl-e in Aare prixe ir. 22 and the primesin Z are prime in any set containing them. More generally, if p A c Band P is prime in B, then p is prime in A.The converse is false--take4 i- 42 c 22. <strong>No</strong>te also that p is prime in A if and only if -p is primein A, so as a result we shall omit discussing the negative primes.There are only two subsets of Z which do not have primes, namely{*!I and { 0, *l}. For any other set A, the least positive integer inA which is greater than one is always prime in A.It is also easy tosee that for any integer n > 1, there is a subset of Z with -n and n asits only primes.in Z, the setFinally, if pl, p2,---p are any n distinct positivenhas exactly 2n primes (n positive primes).<strong>No</strong>te that divisibility istransitive in A so that A serves as a counterexample to the converseof Lemma 1.2. The. Fundwntd Tl~eo~tem 0(1 WlundcIn this section we discuss the primes in nZ, the fundamental theo-rem of arithmetic in nZ and make some observations concerning primes innZ and Z.For the remainder of the paper we use FTA1 to mean thereexists a factorization into a product of primes, FTA2 to mean the fac-torization is unique, and FTA to mean both FTA1 and FTA2.The FTA1 always holds in a subset A of Z of the type consideredearlier (other than A = { *l} and A = [O, el}) and the proof can be pat-terned after that given in [4, p 111. We shall show, however, thatFTA2 does not hold in nZ.FTA does hold; for example take A = {any (positive) prime in Z.There are proper subsets of Z for which the- - ,-p2, -p, p, p2, ' " ' } where p isThere is a simple formula for the primes in n2'(n > 1) while, ofcourse, no such formula i-5 known to exist for the primes in Z.Thiofiem 1':The positive primes in nZ (n > 1) are of the form(kn + i)n for l = 1,2,.;n - 1 and k = 0,1,2;--. All other posiriveintegers in nZ are composite and are of the form kn 2 for k = 1,2;".Proof: Let p c nZ so that p = mn for some rn Z, and suppose thatp is prime in nZ.rn = kn + i, 0 s i < n.Using the division algorithm (in Z) we can write<strong>No</strong>wi = 0 iff p = kn 2 iff n(nZ)p denying the