数理解析研究所講究録998 巻 1997 年 78-92

78

On semi-reduced quadratic
continued fractions and class number$\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{S}_{)}$

Claude LEVESQUE 1\S 1. Introduction
Let $m$ be a square-free integer $>1$ and let. Definequadratic field

be the discriminant of the real$\Delta$$h(\mathrm{Q}(\sqrt{\mathrm{m}})$$\Delta=\{$$4mm,$$0$$\theta=\{$

,

,

if

$\tilde{\omega}$$\sqrt{m}\frac{1}{2}+,\frac{1}{2}\sqrt{m}$$m\equiv 1$(mod 4),if$\ell$$m\equiv 1$$\{$even and 2$3m\equiv$\{m\equiv
2,$2,3$$\int k_{l/2}$

,

.

and

$2|k_{l/2}$,

as a continued fraction:$\lambda_{1}(m)$(resp.

$\lambda_{2}(m)$$x,$ $y,$.

) be the number of solutions of$x^{2}+4yz=\Delta$

with(mod 4),(mod 4),$(\mathrm{m}\mathrm{o}\mathrm{d} 4)(\mathrm{m}\mathrm{o}\mathrm{d} 4),’\ell_{0}\mathrm{d}\mathrm{d}\ell \mathrm{e}\mathrm{V}\mathrm{e}\mathrm{n}$$\overline{\omega}=[k_{0},\overline{k_{1,\ldots,\mathit{1}}k}]$Let$m\equiv 2,3$

1 if

2 if
Write

$\tilde{\omega}=\{$$z\in N=\{0,1,2, \ldots\}$(resp. $x^{2}+4y^{2}=\Delta$ ). Then H. Lu [Lu] proved the following result.

Theorem (Lu). The class number$h_{\Delta}$of$\mathrm{Q}(\sqrt{\mathrm{m}})$is equal to 1 if and only if$\theta+\sum_{i=1}^{l}k\dot{.}=\lambda 1(m)+\lambda_{2}(m)$.

1Written version of a lecture (based on a joint work with E. DUBOIS) given in Kyoto at RIMS
on November 27, 1996, during the symposium Algebraic Number Theory and Related Topics. Let metake this opportunity to express my deepest gratitude to professor Dr. Masanobu KANEKO for hiskind invitation and his support: “Kanekosensei, arigato gozaimasu”. Thanks are also due to professor
Toru NAKAHARA and to professor Hiroki SUMIDA.

79
Problem. Generalize Lu’s result. More precisely, use continued fractions to list
the elements of the set:$\mathrm{t}aX^{22}+bXY+CY$$b^{2}-4ac=\Delta$$a,$ $b,$ $c\in \mathbb{Z},$with$0\leq b<\sqrt{\Delta}\}$of semi-reduced quadratic forms of discriminant
and relatecardinality of this last set, i.e., to the cardinality of the set$\Delta$$\{(a, b, C)\in \mathbb{Z}^{3}$: $\Delta=b^{2}-4ac$ with(resp.

$h_{\Delta}^{+}$$0\leq b<\sqrt{\Delta}\}$$h_{\Delta}$) to the.

\S 2. Preliminaries
A quadratic form$f$is a homogeneous polynomial of the form

$f=f(X, Y)–<a,$$b,$$c>=aX^{2}+bXY+cY^{2}$ , with $a,$$b,$$c\in \mathbb{Z}$,which we may write in matrix form as$f(X, Y)=(XY)( \frac{1}{2}a_{b}$We say that the matrix of$f$$\frac{1}{2}bc)$.

is$M_{f}=($

By definition, the discriminant of$f$$\frac{1}{2}bc$$\frac{1}{2}ba$).

is$\Delta=\Delta_{f;}=b^{2}-4ac$

moreoveris primitive if pgcd $(a, b, c)=1;f$ is definite positive if $\Delta<0,$ $a>0,$ $c>0$ ;and is indefinite if $\Delta>0$ .
Consider$f$$f$$\mathcal{F}_{\Delta}=${primitive quadratic forms of discriminant$\Delta$}.

We need$GL_{2}(\mathbb{Z})=\{$ :$SL_{2}(Z)=\{$ :, ru–st $=\pm 1\}$ ,

$r,$

$s,$

$t,$

$u\in \mathbb{Z}$

$r,$

$s,$

$t,$

$u\in Z$ ,ru–st $=+1\}$ .

80

We can define an action of$A\in GL_{2}(\mathbb{Z})$onby stating that$f$$g=Af$ where $M_{g}=AM_{f}A^{t}$ .

Moreover we say:$f\sim g$$f\approx g$( is equivalent to )
$f$

$g$

( is strictly equivalent to )$f$$g=Af$ for some $A\in GL_{2}(\mathbb{Z})$ ,$\Leftrightarrow$$\Leftrightarrow$$g$$g=Af$ for some $A\in SL_{2}(\mathbb{Z})$ .

(resp.
is
It turns out that the class group (resp. strict class group) of
numberclass
thestrict(resp.
class, its cardinality being denoted the.
of
We are extensively working with the matrices$\mathbb{Q}(\sqrt{\Delta}$$numberh_{\triangle}$$\mathcal{F}_{\Delta}/\approx)$$h_{\Delta}^{+})$$\mathcal{F}_{\Delta}/\sim$$\mathrm{Q}(\sqrt{\Delta})$$E(u)=$ , $A(u)=$ .
We associate to$f$

the quadratic numbers$\omega=\omega(f)=\frac{b+\sqrt{\Delta}}{2|c|}$and$\Omega=\Omega(f)=\frac{b+\sqrt{\triangle}}{2|a|}$and define$\sigma(f)=\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(a)$,.

Gauss defined the right neighbour $R\{f$ ) and the left neighbour $L(f)$ of :$f$

$Rf=A(e)f=<c,$ $-b-2$ ce, $a+be+ce^{2}>$with$e=-\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(C)[\omega(f)]$$Lf=A(E)^{-1}f=<C+bE+aE^{2},$ $-b-2aE,$ $a>$ withWe say that $f=<a,$$b,$$c>\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\triangle>0$

$\{$

(i)
(ii)

,

$E=-\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(a)[\Omega(f)]$.

is reduced if$0<b<\sqrt{\Delta}$,$\sqrt{\triangle}-b<2|c|<\sqrt{\triangle}+b$,

which is equivalent to saying
$\{$

(i)
(ii)

$0<b<\sqrt{\Delta}$,$\sqrt{\Delta}-b<2|a|<\sqrt{\triangle}+b$.

Gauss showed how to calculate the class number . With the continued fraction
as aalgorithm, write the finite set of reduced quadratic forms of discriminant$h_{\Delta}^{+}$$\triangle>0$81
disjoint cycles, by picking up a reduced quadratic form $f=<a,$union oftaking the continued fraction of$h_{\Delta}$$\omega_{0}=\omega(f)=\frac{b+\sqrt{\Delta}}{2|c|}=\frac{P_{0}+\sqrt{\triangle}}{2Q_{0}}$

$b,$

$c>$

and

,

i.e., for $i\geq 0$ ,

.

$\{$

,

$\omega_{i}$$=$$\frac{P_{i}+\sqrt{\Delta}}{2Q_{i}}$$P_{+1}$$=$$2k_{i}Q_{i^{-}}P_{i}$$Q_{i+1}$$=$$\frac{\triangle-P_{i+1}^{2}}{4Q_{i}}$with$k_{i}=[_{\iota’}v_{i}]$,,where $[]$ is the greatest integer function. Hence

It is well known that is reduced if and only if
is a reduced quadratic number(i.e., the continued fraction expansion ofis purely periodic).
Let
if is even,
if is odd.
The cycle of is,$f$$\omega(f)$$\omega(f)$$\ell$$\ell$$2\ell$$\ell$$\ell=\sim\{$$f$$[f_{1},$$f_{2},$$\ldots,$$f_{l}’\{$

where$f=f_{1}=<a,$and where for $i=1,$$\ell-1\sim$$\ldots,$$f_{i+1}$$b,$$c>=<\sigma(f)Q0,$ $P1,$ $-\sigma(f)Q1>$,$=$$R(f_{\dot{\iota}})$$=$$\langle(-1)^{i}\sigma(f)Qi,$ $P_{i}+1,$ $(-1)i+1\sigma(f)Q_{i+}1\rangle$$=$$A_{i}f_{i}$with$A_{i}=A(\sigma(f_{i})k_{i})=A((-1)^{i-1}\sigma(f)k_{i}\mathrm{I}\cdot$

In symbols,$f_{+1}\dot{.}$$=$$R.f_{i}$$f_{j}$$=$$Lf_{j+1}$

for$1\leq i\leq\ell\sim$

with

for

$0\leq j<\ell\sim$

with

,

$f_{1}=Rf_{\overline{\ell}}$$\{$$f_{\overline{\ell}}=Lf_{1}$

,

82

.

and $=[\omega(f.)]$ .
(resp.
is the number of cycles. Moreover
Fact:
form
quadratic
any
reducedforlength of the continued fraction of$k$$h_{\Delta}^{+}=h_{\Delta}$$h_{\Delta}^{+}$) if theis odd (resp.

$h_{\Delta}^{+}=2h_{\Delta}$$\omega(f)$$f$even).
See section 7 for examples of calculations of cycles.

\S 3. Semi-reduced forms
A quadratic form $f=<a,$$b,$$c>\mathrm{o}\mathrm{f}$discriminant$0\leq b<\sqrt{\triangle}$$\triangle$

is semi-reduced if,

and is said to be intermediate if est semi-reduced without being reduced.
In each of the following examples, we give the list of semi-reduced quadratic forms,
the reduced forms being written in boldface, the intermediate forms appearing in romanstyle.
Example 3.1. Let $m=14$ . There are 24 semi-reduced quadratic forms of discrim:
inant
$f$

$f$

$\triangle=56$$<1,0,$ $-14>$$<-1,0,14>$$<2,0,$ $-7>$$<-2,0,7>$$<7,0,$ $-2>$$<1,2,$ $-13>$$<-1,2,1\mathrm{s}>$$<14,0,$ $-1>$$<-14,0,1>$$<13,2,$ $-1>$$<1,4,$ $-10>$$<-1,4,10>$$<2,4,$ $-5>$$<5,4,$ $-2>$$<10,4,$ $-1>$$<-10,4,1>$$<1,6,$ $-5>$$<-2,4,5>$$<-1,6,5>$$<5,6,$ $-1>$$<-7,0,2>$$<-13,2,1>$$<-5,4,2>$$<-5,6,1>$

Example 3.2. Let $m=19$ . There are 36 semi-reduced quadratic forms of discriminant $\Delta=76$ :$19>$$<19,0,$ $-1>$$<-19,0,1>$$<1,2,$ $-18>$$<-1,2,18>$$<2,2,$ $-9>$$<-2,2,9>$$<3,2,$ $-6>$$<-3,2,6>$$<6,2,$ $-3>$$<9,2,$ $-2>$$<-9,2,2>$$<18,2,$ $-1>$$<-18,2,1>$$<1,4,$ $-15>$$<3,4,$ $-5>$$<-3,4,5>$$<-1,6,10>$$<-10,6,1>$$<5,4,$ $-3>$$<-5,4,3>$$<-2,6,5>$$<-1,8,3>$$<15,4,$ $-1>$$<-6,2,3>$$<-1,4,15>$$<-15,4,1>$$<-5,6,2>$$<-3,8,1>$$<1,0,$ $-19>$$<1,6,$ $-10>$$<10,6,$ $-1>$$<-1,$$\mathrm{o},$$<2,6,$ $-5>$$<1,8,$ $-3>$$<5,6,$ $-2>$$<3,8,$ $-1>$

Example 3.3. Let $m=26$ . There are 36 semi-reduced quadratic forms of discriminant $\Delta=104$ :$<1,0,$ $-26>$$<26,0,$ $-1>$$<25,2,$ $-1>$$<11,4,$ $-2>$$<17,6,$ $-1>$$<5,8,$ $-2>$$26>$$<2,0,$ $-13>$$<-26,0,1>$$<-25,2,1>$$<-11,4,2>$$<-17,6,1>$$<-5,8,2>$$<1,2,$ $-25>$$<-1,$$\mathrm{o},$$<1,4,$ $-22>$$<22,4,$ $-1>$$<1,8,$ $-10>$$<10,8,$ $-1>$$13>$$<13,0,$ $-2>$$<-1,2,25>$$<-1,4,22>$$<-22,4,1>$$<-1,8,10>$$<-10,8,1>$$<5,2,$ $-5>$$<-2,$$\mathrm{o},$$<2,4,$ $-11>$$<1,6,$ $-17>$$<2,8,$ $-5>$$<1,10,$ $-1>$$<-13,0,2>$$<-5,2,5>$$<-2,4,11>$$<-1,6,17>$$<-2,8,5>$$<-1,10,1>$

83
Example 3.4. Let $m=33$ . There are 20 semi-reduced quadratic forms of discriminant $\Delta=33$ :$<1,1,$ $-8>$$<-1,1,8>$$<2,1,$ $-4>$$<-2,1,4>$$<4,1,$ $-2>$$<-4,1,2>$$<2,3,$ $-3>$$<8,1,$ $-1>$$<1,3,$ $-6>$$<-1,3,6>$$<-2,3,3>$$<-8,1,1>$$<3,3,$ $-2>$$<-3,3,2>$$<6,3,$ $-1>$$<-6,3,1>$$<1,5,$ $-2>$$<-1,5,2>$$<2,5,$ $-1>$$<-2,5,1>$

Example 3.5. Let $m=35$ . There are 40 semi-reduced quadratic forms of discriminant $\triangle=140$ :$<19,8,$ $-1>$$<-19,8,1>$$<-1,8,19>$$<1,0,$ $-35>$$<-1,0,$$<7,0,$ $-5>$$<-7,0,5>$$<2,2,$ $-17>$$<35,0,$ $-1>$$<-17,2,2>$$<1,6,$ $-26>$$<-13,6,2>$$<5,10,$ $-2>$$<34,2,$ $-1>$$<-1,2,34>$$<17,2,$ $-2>$$<-31,4,1>$$<13,6,$ $-2>$$<-2,10,5>$$<1,10,$ $-10>$$\mathrm{s}5>$$<-2,2,17>$$<-1,6,26>$$<26,6,$ $-1>$$<-5,10,2>$$<-1,10,10>$$<1,8,$ $-19>$$<5,0,$ $-7>$$<-5,0,7>$$<-35,0,1>$$<1,4,$ $-31>$$<1,2,$ $-34>$$<-34,2,1>$$<2,6,$ $-13>$$<31,4,$ $-1>$$<-26,6,1>$$<10,10,$ $-1>$$<2,10,$ $-5>$$<-1,4,31>$$<-2,6,13>$$<-10,10,1>$

Let $f=<a,$a reduced form of discriminant such that $e=\sigma(f)[\omega(f)]$ .
Consider the quadratic forms associated to and given by the following two cases (thefirst may be empty):$b,$$c>\mathrm{b}\mathrm{e}$$\Delta$$f$Case (i)Case (ii)$E(u)f=<a+bu+Cu^{2},$$b+\mathrm{W}\mathrm{i}\mathrm{t}\mathrm{h}2_{Cu,c>}$$A(u)f=<C,$$-b-2Cu\mathrm{W}\mathrm{i}\mathrm{t}\mathrm{h}’ a+bu+cu^{2}>$$\frac{b}{2|c|}\leq\sigma(f)u\leq\sigma(f)e=[\omega(f)]$$1 \leq\sigma(f)u\leq\frac{b}{2|c|}$We can show that the quadratic forms of Cases (i) and (ii) are all semi-reduced(i.e., are either reduced or intermediate) and that the intermediate forms $<a’,$Cases (i) and (ii) respectively verify the following properties:$b’,$ $d>\mathrm{o}\mathrm{f}$$a’d<0$ ,Case (i)$\{$$|a’|>|c’|$,$c>=Rf$ ,$R<a’,$ $b”,$$a’d<0$ ,Case (ii)$\{$$|a’|<|d|$ ,$L<a’,$ $b’,$ $d>=f$.

These last properties characterize intermediate quadratic forms. We can show that84
anintermediate forms come from reduced forms. More precisely, let $g=<a’,$intermediate quadratic form. Then the following properties hold true:
both reduced(i) If $|a’|>|d|_{:}$. then the forms $f_{1}=Rg$ and $f=Lf_{1}=<a,$and$b’,$ $d>\mathrm{b}\mathrm{e}$$b,$$c>\mathrm{a}\mathrm{r}\mathrm{e}$$g=E(u)f=<a+bu+cu^{2},$ $b+2cu,$ $c>$

wi.t$\mathrm{h}$$1 \leq\sigma(f)u\leq\frac{b}{2|c|}$(ii) If $|a’|<|d|$ , then the form $f=Lg=<a,$

$b,$

.

$c>\mathrm{i}\mathrm{s}$reduced and$g=A(u)f=<c,$ $-b-2Cu,$ $a+bu+cu^{2}>$

with

.

$\frac{b}{2|c|}\leq\sigma(f)u\leq\sigma(f)e=[\omega(f)]$We can now state our first result.
Theorem. (1) For each reduced form $f=<a,$forms associated to and defined by$b,$$c>of$ discriminant$\triangle$, the quadratic$f$,$E(u)f$with$1 \leq\sigma(f)u\leq\frac{b}{2|c|}$$A(u)f$with$\frac{b}{2|\mathrm{c}|}\leq\sigma(f)u\leq[\omega(f)]$

$\{$

,

are all different from one another and form a set $I(f)$ of cardinality$[\omega(f)]$$\neq I(f)=\{$$1+[\omega(f)]$ifif$(2c),\gamma b$$(2_{C})|b$,.

(2) If and are two different reduced forms, then $I(f)\cap I(g)=\phi$ .
(3) Moreover the (disjoint) union of the $I(f)^{J}s$ when runs through the set Red
of reduced forms is a set equal to the set of semi-reduced forms and is of cardinality
$f$

$g$

$f$

2
where

$\tau$$(_{\mathrm{t}\equiv\Delta(}0 \leq|<\sqrt{\Delta}\tau\sum_{)mod2}(\frac{\Delta-t^{2}}{4}\mathrm{I}1$is the number of divisors function.

,$(\Delta)$

85\S 5. The palindrome level$\alpha$Suppose that the continued fraction of a quadratic number
is reduced):$\alpha$is purely periodic (i.e.

$\alpha=[\overline{k_{1},k_{2},\ldots,k_{\ell}}]$with$\ell$

minimal. We associate to$\alpha$the word$w(\alpha)=k_{1}k2\cdots k_{\ell}$and say that it is defined up to a cyclic permutation (of order ).
Recall that a palindrome $(KA I BUN)$ is a word which once read ffom left to rightor from right to left is the same.
$\ell$

Examples. In french, english, german and japanese:(i) ELU PAR CETTE CRAPULE(ii) NAME NO ONE MAN(iii) EIN NEGER MIT GAZELLE ZAGT IM REGEN NIE(iv)

6$\langle$$l\mathrm{f}^{f_{\mathrm{f}}}\emptyset$$\mathit{0})$$\S\not\equiv \mathrm{f}\not\equiv\emptyset\searrow\iota_{\sim}^{arrow}$$\mathrm{t}\mathit{0})\mathit{0})fX$$\mathrm{k}^{>}\emptyset\searrow\iotaarrow\phi\sim \mathrm{g}\searrow f\not\equiv$

(v)
(vi)

$\tau.\mathrm{g}k\emptyset \mathrm{b}$$f_{-}^{\sim}t\tau\varphi$

A$\emptyset\searrow k^{\mathrm{Y}}$efa$\langle$$X\xi$ )6$\ovalbox{\tt\small REJECT} \mathrm{S}\vee)\mathrm{b}\emptyset \mathrm{k}^{\mathrm{Y}}\mathrm{g}\tau$$\iota_{\mathit{1}^{1}}\mathrm{S}\ovalbox{\tt\small REJECT}|_{\vee}1_{\mathit{1}^{1}}\mathrm{b}$‘$\mathrm{f}l\mathrm{y}\gamma_{\mathrm{c}}\tau$When is odd, we say that has 1 (resp. ) central element if
is (resp.
is not) a palindrome, the central element being the center of. When is even,
we say a has 2 central elements (resp.
central element) if after an eventual cyclic
permutation,is (resp. is not) the concatenation of a palindrome of length $\ell-1$
and of a palindrome of length 1, the two central elements being the two centers of theconcatenated palindromes.
Definition. The palindrome level $s=s(\alpha)\in\{0,1,2\}$ of a reduced quadratic
number is equal to the number of even central elements.
In examples 7.1 to 7.5, is respectively equal to 2, 1, 1, $0,0$ .
$\ell$$0$$\alpha$$w(\alpha)$$w(\alpha)$$\ell$$0$$w(\alpha)$$s$\S 6. The main results
We can now relate the class numberofto the cardinality of the set ofsemi-reduced forms by writing this cardinality in terms of the partial quotients of nonequivalent reduced quadratic numbers.
$h_{\Delta}$$\mathrm{Q}(\sqrt{\Delta})$

86
Theorem. Let$\Delta$be the disc$7\dot{\tau}minant$$0$$g=\{$$2^{r+1}$$2^{r+2}$

if
if
if

of a real quadratic field$\Delta\equiv 1$$\Delta\equiv 4$$\Delta\equiv 8$$\mathrm{Q}(\sqrt{\Delta})$. Put(mod 4),ou 12 (mod 16),
(mod 16),

where is the number of odd primes dividing . Then the class numberofis equal to
if and only if there exist reduced quadratic numbers
associated toquadratic forms non equivalent to one another such that the numberof all semi-reduced forms of discriminant is$\Delta$$r$$h_{0}$$h_{\Delta}$$\alpha^{(1)},$$h_{0}$$\mathrm{Q}(\sqrt{\Delta})$$\alpha^{(h_{0})}$$\alpha^{(2)},$$\ldots,$$h_{0}$$\Delta$2
where for $j=1,$$h_{0}$$\ldots,$$(s_{\Delta}+ \sum_{1j=}^{h_{0}}\sum_{t=1}^{l_{\mathrm{j}}}k_{t}^{(}j$),$\mathrm{I}$,$\alpha^{(j)}=[\overline{k_{1}(j),k_{2}(j),\ldots,k^{(}j)}]l_{j}$with minimal, and where the palindrome level
palindrome levels of the’s, is equal to$\ell_{j}$$s_{\Delta}$

of$\Delta$, defined as the sum of all the$\alpha^{\langle j)}$$s_{\Delta}= \sum_{1j=}S(\alpha)h0(j)=\frac{g}{2}$Theorem. Let$\Delta$.

be the discriminant of a real quadratic field$0$$g_{1}=\{$$2^{r}$$2^{r+1}$

if
if
if

$\Delta\equiv 1$$\Delta\equiv 4$$\Delta\equiv 8$$\mathrm{Q}(\sqrt{\Delta})$. Put(mod 4),ou 12 (mod 16),
(mod 16),

where is the number of odd primes dividing . Then the class numberof),is equal to
and
there
only
exist
reducedquadratic,
numbersif
if
associated toquadratic $f_{ormS}.$
another such thateq.uiva.lent- to$\Delta$$r$$h_{1}$non.

$h_{1}$$\#\{(A, B, c)\in N^{3} :where for $j=1,$$\ldots$,$h_{1}$$h_{\Delta}$$\beta^{(1)},$$h_{1}$$\mathrm{Q}(\sqrt{\Delta})$$\beta^{(2}$$\beta^{(h_{1})}$$\ldots$

one.

\Delta=B^{2}+4AC\}=s_{\Delta}+\sum_{j=1}^{h_{1}}\sum^{j}k^{(j)}t=\ell 1t$’,$\beta^{(j)}=[\overline{k_{1’ 2}^{(j)}k\langle j),\ldots,k(j)}]l_{\mathrm{j}}$with minimal, and where the palindrome level
palindrome levels of the’s,$\ell_{j}$$\beta^{(j)}$$s_{\Delta}$

of

$is$

$s_{\Delta}= \sum_{=j1}^{h_{1}}s(\beta \mathrm{t}j))=g_{1}$.

$\Delta_{f}$defined as the sum of all the

87\S 7. Five examples
One has a chance of verifying the last theorems ith the following examples.
$\dot{\mathrm{w}}$

Example 7.1. Let $m=14$ , so $\Delta=56$ . Here:isfraction expansion of$\omega=\frac{4+\sqrt{56}}{2\cdot 2}$$h_{\Delta}^{+}=2$

and$h_{\Delta}=1$. The continued$[\overline{2,1,6,1}]$$\omega=\frac{2+2\sqrt{10}}{2\cdot 3}rightarrow$

Since $w(\omega)=1,6$ , 1, 2, we have $s(\omega)=2$ . There are two cycles of reduced forms$arrow-$(in boldface) to which are attached the intermediate forms:

88
Example 7.2. Let $m=19$ , so $\Delta=76$ . Here:ishaction expansion of$\omega=\frac{4+\sqrt{76}}{2\cdot 5}$$h_{\Delta}^{+}=2$

and$h_{\Delta}=1$. The continued$[\overline{1,3,1,2,8,2}]$$\omega=\frac{4+\sqrt{76}}{2\cdot 5}rightarrow$

Since $w(\omega)=1,2,8$ , 2, 1, 3, we have $s(\omega)=1$ . There are two cycles of reduced– –forms (in boldface) to which are attached the intermediate forms:

89
Example 7.3. Let $m=26$ , so $\Delta=104$ . Herefraction expansion ofis:$\omega_{1}=\frac{8+\sqrt{124}}{2\cdot 2}$$h_{\Delta}^{+}=h_{\Delta}=2$. The continued$[\overline{4,1,1}]$$\omega_{1}=\frac{8+\sqrt{104}}{2\cdot 2}rightarrow$

Since $w(\omega_{1})=1,4,1\sim$’ we have $s(\omega_{1})=1$ .
The continued ffaction expansion of$\omega_{2}=\frac{10+\sqrt{124}}{2\cdot 1}$

is$[\overline{10}]$:$\omega_{2}=\frac{10+\sqrt{104}}{2\cdot 1}rightarrow$

Since $w(\omega_{2})=$$10$, we have $s(\omega_{2})=1$ . There are two cycles of reduced forms (in$\sim$

boldface) to which are attached the intermediate forms:$<-11,4,2>$$<-13,0,2>$$<2,0,$ $-13>$$<2,4,$ $-11>$$<11,4,$ $-2>$$<13,0,$ $-2>$$<-2,0,13>$$<-2,4,11>$

90$<-10,8,1>$$<-17,6,1>$$<-22,4,1>$$<-25,2,1>$$<-26,0,1>$$<10,8,$ $-1>$$<17,6,$ $-1>$$<22,4,$ $-1>$$<25,2,$ $-1>$$<26,0,$ $-1>$$<1,0,$ $-26>$$<-1,0,26>$$<-1,2,25>$$<-1,4,22>$$<-1,6,17>$$<-1,8,10>$$<1,2,$ $-25>$$<1,4,$ $-22>$$<1,6,$ $-17>$$<1,8,$ $-10>$

Example 7.4. Let $m=33$ , so $\Delta=33$ . Here:isfraction expansion of$h_{\Delta}^{+}=2$

and$h_{\Delta}=1$. The continued$[\overline{5,2,1,2}]$$\omega=\frac{5+\sqrt{33}}{2\cdot 1}$$\omega=\frac{5+\sqrt{33}}{2\cdot 1}rightarrow$

Since $w(\omega)=2,1,2,5$ , we have $s(\omega)=0$ . There are two cycles of reduced forms$-arrow$(in boldface) to which are attached the intermediate forms:$<-2,3,3>$$\Rightarrow$$arrow$$<1,5,$ $-2>$$arrow$$<-2,1,4>$$\Downarrow$$\Uparrow$$<4,1,$ $-2>$$<3,3,$ $-2>$$\Leftarrow$$<-2,5,11>$$<-6,3,1>$$<-8,1,1>$$<1,1,$ $-8>$$<1,3,$ $-6>$

91

Example 7.5. Let $m=41$ , so $\Delta=41$ . We will see
is:fraction expansion of$\omega=\frac{5+\sqrt{41}}{2\cdot 1}$$h_{\Delta}^{+}=h_{\triangle}=1$. The continued$[\overline{5,1,2,2,1}]$$\omega=\frac{5+\sqrt{41}}{2\cdot 1}rightarrow$

Since $w(\omega)=2,1,5$ , 1, 2, we have $s(\omega)=0$ . There is only one cycle of reduced forms–(in boldface) to which are attached the intermediate forms.

$<-8,3,1>$$<-10,1,1>$$<1,1,$ $-10>$$<2,1,$ $-5>$$<1,3,$ $-8>$$<-5,1,2>$$<5,1,$ $-2>$$<-2,1,5>$$<8,3,$ $-1>$$<10,1,$ $-1>$$<-1,1,10>$$<-1,3,8>$

92

References[B] Buell, D. A., Binary quadratic forms, Springer-Verlag New York Inc., (1989),$\mathrm{x}+247$

pages.

[D-L] Dubois, E. and Levesque, C., Formes quadratiques semi-r\’eduites, fractions
continues et nombre de classes, to be submitted to L’Enseignement Math\’ematique.

[F] Flath, D. E., Introduction to Il um ber theory, John Wiley &Sons, (1989),
pages.

$\mathrm{x}\mathrm{i}\mathrm{i}+212$[Lu] Lu, H., On the class number of real quadratic fields, Scientia Sinica II (special
number, 1979), 118-130.
[M] Mollin, R., Quadratics, CRC Press, Boca Raton,$\mathrm{x}\mathrm{x}+387$

pages.

Claude LEVESQUE
D\’epartement de math\’ematiques et de statistique
Universit\’e Laval
Qu\’ebec
CanadaG1K$7\mathrm{P}4$