Class Number One Criteria :For Quadratic Fields. I

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No. ] Proc. Japan Acad.,,63, Ser. A (987) Class Number One Criteria :For Quadratic Fields. I Real By R. A. MOLLIN Mathematics Department, University of Calgary, Calgary, Alberta, Canada, TN N (Communicated by Shokichi IYANAGA, M. $. A., April 3, 987) In [] we established criteria for Q(/-) to have class number, h(n), equal to one when n=m+l is square-free. Portions of this result were rediscovered by Yokoi [] and Louboutin [], both of whom also found similar criteria for square-free integers of the form n--m+. It is the purpose of this paper to generalize all of the above by providing criteria for h(n)= for a positive square-free integer n-- (mod ), under a certain assumption, which is satisfied (among others) by Richaud-Degert (R-D) types described below. One of these criteria is that x +x + (n- )/ is equal to a prime for all integers x e (, (/n--)/). This is the exact real quadratic field analogue of" h(--p)- if and only if x-x + (p + )/ is prime for all integers x e [, (p-7)/] where p--3 (mod ) is prime and p7. This was proved by Rabinowitsch [0] (see also [], [], and [3]). We apply the criteria to real quadratic fields of narrow R-D type i.e., those n=m/r where Ir e {, }, n=/=. We also observe that when n=m + the existence of exactly six quadratic fields with h(n)= can be established by the same method used by Mollin and Williams in [9] to verify a similar fact for the case n=m+. The following notation is in force throughout the paper. For the field Q(/-) we denote the fundamental unit by (T/U/--)/a, a- if n--i (mod ), and a otherwise. Moreover N((T + U/-)/a) where N denotes the norm from Q(/-) to Q. For convenience sake we let A ----((T/a) --g--)/u. Firs we sae he ollowing resul which we will need or he firs main heorem. The proo o he ollowing can be found in [] (see also [8). Lemma. Le$ n be a square-free positive integer. If h(n)=l then p is inert in Q(/--) for all primes p<a. The converse of this Lemma is clearly false. For example, if n--3 then a----l, T--3, U=6, and /-- so A--68/36<. However, h(3)-. However, the converse does hold under certain circumstances, as the following main result illustrates. Theorem. Let n--:l (mod ) be a positive square-free integer, such that (/n-)/<_a. Then the following are equivalent. () h(n)---

R.A. V0LLIN [Vol. 63 (A), () p is inert in Q(/-) for all primes pa (3) f(x)=-x+x+(n-)/o (mod p) for all integers x and primes p satisfying O<xp<(/n-)/; () f(x)is equal to a prime forallintegersxsuchthat l<x(/-l)/. Proof. () follows from () by the Lemma; (note that in this case (/n-- )/_A is not required). Assume now that () holds. If f(x) ----0 (mod p) or some O<x<p<(/n-)/ then n=(x-) (mod p) whence p is not inert in Q(/W). By () this forces (/n-)/>a, contradicting the hypothesis. Thus () implies (3). Assume (3) holds. If (n.-)/ is composite, but not the square of a prime, then there exists a prime p dividing (n- )/ such that f()-0 (mod p) with o<<p<(/n-)/. This contradicts (3). Hence or some prime p we must have that (n-)/=p or p. Suppose that there are primes p, and p. (not necessarily distinct) such i pp that _(n- f(x)=_o (modp,p)or some integer x with l<x<(/n-)/, )/ then x +x + (n- )/_ (n- ) / whence x_, a contradiction. Therefore, without loss of generality we may assume that p, <(/n--)/. If p, divides x then p, divides (n-l)/; whence p,=p. However we have that p=p, gx<(/--l)/_p, a contradiction. Hence, in consideration of the congruence f(x)=_o (mod p,) we may assume without loss of generality that 0<x<p,. Hence, we have f(x)=0 (modp,) with O<x<p,<(v/n-)/ which cntradicts (3). Thus (3) implies (). Finally assume that () holds. If h(n)> then by [3, Propositions 3 and, p. 6] there exist an integer x and a prime p such that 0_x <p _(/---- )/ and both" (a) (b) N((x----/W)/)_O (modp) and there does not exist an integer k such that N(x + kp- - /W)/ p. From (a) it ollows that x + x + (n-- )/-- 0 (mod p). Therefore, if x(/n-)/ then, by (), -x+x+(n-)/-p. However xp _(/-)/; whence p--x(-- x)+(n--)/p(--p)+p=p, a contradiction. Hence x--0 or. Therefore p divides (n--l)/; whence f(p)- p(--p+l+(n--)/p). I p(/n-)/ then () implies that f(p)=p. Thus p-- (/n: )/, a contradiction. Hence p= (/n- )/. Setting k in (b) yields that" pln(p+_l--/w)/-(p+p+l--n)/- p, a contradiction. This secures the result. Q.E.D. The ollowing special case of the Theorem for certain R-D type was proved in []. It was also rediscovered by Yokoi [] and Louboutin []. See also [7]. Corollary. If n--+l is square-free where either n is composite or is composite then h(n). If n--q+ where n and q are primes then the following are equivalent" () h(n)= () p is inert in Q(/W) for all primes pq

No. ] Class Number of Real Quadratic Fields. I 3 (3) f(x)----x+x+qo (modp) for all integers x and primes p such that Oxpq () f(x) equals a prime for all x with lxq. Proof. By [] and [] T= and U=. Moreover, =--, (/n--)/ =l and A=l. Thus the hypothesis of the theorem is satisfied. Q.E.D. S. Chowla conjectured that ifp=m+ is prime with m6 then h(p). Thus Crollary reduces the conjecture to the case where m=q, q 3 prime. This exhausts the algebraic techniques (see []). Using analytic techniques and the generalized Riemann hypothesis, Mollin and Williams proved the Chowla conjecture in [9]. We now turn to another interesting consequence of the Theorem. The following R-D types were also considered by Yokoi [] and Louboutin []. Both o these authors results ollow as a special case o the ollwing. Corollary. Let n=m + be square-free. Then h(n)l unless n=p+ l where p is prime. In this case the following are equivalent" () h(n)=l; if n=m () q is inert in Q(/W) for all primes q + {: - if n=m- (3) f(x)=-x+x+po (mod q) for all integers x and primes q sarislying O x q /-- () f(x) is equal to a prime for all integers x satisfying lx/. Proof. By [] and [] T=m and U=I. An easy check shows that (/----)/_A. Thus the hypothesis of the Theorem is satisfied, and the equivalence ()-() is secured. It remains to show that h(n)l unless n---m + p + where p is prime. Suppose that (n--)/ is not prime and h(n)=l. Then (3) of the Theorem implies, by the same reasoning as in the proof o the Theorem, that (n--)/=p or some prime p. Therefore m--p= (respectively m--p= 3) when n=m-- (respectively n=m+). In the ormer case m + p-- is orced, contradicting m 3 and in the latter case m- p-- 3 is forced, contradicting m. This shows that n=p + or some prime p when h(n). Q.E.D. Remark. In [] Yoki conjectured that h(n)l when n=q+ is square-ree with q 7 prime. Under the assumption of the generalized Riemann hypothesis this conjecture llows in the same ashion as did the analogous Chwla conjecture proved by Mollin and Williams in [9]. Remark. Suppose that n=p-t-l=m+ where p is a prime and m is a psitive integer. If s/- is an odd prime then p=t (mds) or 0_t s. I there exists an integer u0 such that l+t--(u--) (mod s) then f(u)=--u+u+p--o (mods) where Ous/-. This violates condition (3) of Crollary. Hence h(n). (See [6] or connections with generalized Fibonacci primitive rots.) The ollowing Table illustrates Cor.llaries -. We list the r= case

R.A. MOLLIN [Vol. 63 (A), only up to m=6 since we know by Remark that h(n)>l for m>6. Similarly we list the r= only up to m= 7. For r=-- with h(n)= it is unlikely that any ether such n exist than these listed in the Table. Table. n=m+r m r n h(n) 6 8 0 6 0 6 7 9 3 7 9 309 - - - - 37 6 0 97 7 0 8 677 9 3 8 73 9 93 77 37 977 3 3 All class numbers are taken rm []. In a subsequent work we will look at wide R-D types in detail. Acknowledgement. This research was supported by both N.S.E.R.C. Canada Grant #A88 and an I. W. Killam Award held at the University of Calgary in 986. References [] [] [3] [] [] [7] R. G. Ayoub and S. Chowla: On Euler s Polynomial. J. Number Theo.ry, 3, 3.- (98). G. Degert: ber die. Bestimmung der Grudeinheit gewisser reell-quadratischer ZahlkSrper. Abh. Math. Sem. Univ. Hamburg,, 9-97 (98). M. Kutsuna: On a criterion for the class number of a quadratic number field to be one.. Nagoya Math. J., 79, 3-9 (980). S. Louboutin: Critres des principalit et minoration des. hombres de classes l idaux des corps quadratiques rels l aide de la t.horie des fractions continues (preprint). R. A. Mollin: Necessary and sufficient conditions for the class numb.e.r of a real quadratic field t.o be one, and a conjecture of S. Chowla (to. appear in Prec. Amer. Math. Soc.). ----: Generalized Fibonacci primitive roots, and class numbers of real quadratic fields (to appear in The Fibonacci Quarterly). Dio.phantine equations and class numbers. J. Number Theory,, 7-9 (986).

No. ] Class Number of Real Quadratic Fields. I [8] [9] [0] [] [] [3] [] [] R. A. Mollin: On the insolubility of a class of diophantine equations and the nontriviality of the class numbers of related real quadratic fields of Richaud-Degert type (to appear in Nagoya Math. J.). R. A. Mollin and H. C. Williams: A conjecture of S. Chowla via the. generalized Riemann hypothesis (to appear in Proc. Amer. Math. Soc.). G. Rabinowitsch: Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen ZahlkSrpern. J. Rein Angew Math.,, 3-6 (93). C. Richaud" Sur la. rsolutio.n des quations x--ay--+-l. Atti. Accad. Pontif. Nuovi Lincei, 77-8 (866). R. Sasaki: On a lower bound for the class number of an imaginary quadratic field. Proc. Japan Acad., A, 37-39 (986). H. M. Stark: A complete determination of the complex quadratic fields of classnumber one. Michigan Math. J.,, -7 (967). H. Wada: A table of ideal class numbers of real quadratic fields. Kokyuroku in Math., 0, Sophia University, Tokyo. (98). H. Yokoi: Class-number one problem for certain kinds of real quadratic fields (preprint series 7, Nagoya University (986)).

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