primality of p. Thus if p is an nZ-prime, then p = (kn + ¥i) fori = 1,2,- - -, n - 1.ThiOJLem 2: For any integer n 3 2, the FTA2 does not hold in n2.Proof: By Theorem 1, iz2(n + I ) ~ is composite in nZ and is notuniquely factorable sincen2(n + I ) ~ = n . [n(n + 1l2] andn2(n + I ) ~ = [n(n + l)] [n(n + l)]are two differenr prime facrorizations in nZ.The following rheorem identifies which composires in nZ are and whichare not uniquely factorable .in nZ.First we observe that if x is com-posite in nZ, we have x = kn7 by Theorem 1. We then note that x can bewritten as x = kiz2 = k$where m 2 2 and n 1 k,.mT h e m 3: Let x be an 7x2-composite written x = kn where m  2and n 1 k.prime in Z.Proof:Then x is uniquely factorable if and only if k = 1 or k isLet k = pla1p2a2. . .ppap be the unique prime factorizationof k in 2. Observing that k = 1 is a trivial case, we go on to thecases when k # 1.Case 1: r =1.m- 1(a) If a, = 1 then x is uniquely factorable as x = n (np).(b) If a, > 2, then x is not uniquely factorable sincem-1x = 71 (nplal )are two different prime factorizations of x in n2.cd~e2: P a 2.Then x is not uniquely factorable since- -- m-2* x = n (nplal )(np2'2. .p:r)= nm-l(nplal.. .ppa')are two different prime factorizations of x in nZ.Theorem 1 allows us to make the following obser-;2:~0:.~primes in nZ and in 2; some of these will be disc.:sse=concerningi: -.ore de~ail inthe next section.(1) There are infinitely many primes in each ?&Z, as there are inz.(2) There are not arbitrarily large gaps between primes in nZ asthere are in Z. In fact, the gap between primes in È iseither 0 or 1.(3) There are infinitely many twin primes (primes separated byonly one composite) in each nZ.Moreover, the twin primes innZ have the form: (kn + (n - 1)) n and ((k + DM + 1) . nfor k = 0,1,2, .-.(4) Given any positive integer n, however large, nZ contains ?a - 1consecutive primes.primes is the sequence 2, 3.)3. The AflA,t/vn&tic Function n [ XI(In Z the largest sequence of consecutiveL ~ TT (x) denote rhe number of posirive primes in A which are less thanAor equal to x e A, Since the primes and composites in iiZ are so nicelydistributed, we are able to obtain an explicit formula for this function.In the following theorem, [x] means the greatest integer (in Z) which isless than or equal to x.Theokem 4:If x = kn e. nZ, thenn - 1(kn) = k- [k/n] =-nz(kr ) + r ,where in the last equality k = nq + r and 0 2 r < 72, q = [k/x].Proof',The number of positive primes in nZ which are less than orequal to kn is equal to T - C whereandT = the total number of p6sftive integers in ?22 which are .< kn,C = the number of positive composites in ni! which are  kn.It is clear that T = k.are of the form kn2,As we saw in Theorem 1, the composites in nZ<strong>No</strong>w the composites ip nZ which are less than orequal to kn are : n2, 2n2, 0 * , [k/iz] n2. Thus, C = [k/n]-In Z, the ratio n(x)/x can be interpreted in two ways':(1) as the number of positive primes which are  x compared to x, or(2) as the number of positive primes which are 5 x compared tothe number of positive integers which are < x.