#A16
INTEGERS 11B (2011)
A NEW ALGORITHM OF CONTINUED FRACTIONS RELATED
TO REAL ALGEBRAIC NUMBER FIELDS OF DEGREE ≤ 5
Jun-ichi Tamura
3-3-7-307 Azamino Aoba-ku Yokohama, Japan
jtamura@tsuda.ac.jp
Shin-ichi Yasutomi
General Education, Suzuka National College of Technology, Shiroko, Japan
yasutomi@genl.suzuka-ct.ac.jp
Received: 9/15/11, Revised: 3/10/11, Accepted: 7/15/11, Published: 12/2/11
Abstract
In an earlier paper, we introduced a new algorithm which is something like the
modified Jacobi-Perron algorithm, and gave some computer experiments by which
we can expect that the expansion obtained by our algorithm for α = (α1 , . . . , αs ) ∈
K s (with some natural conditions on α) becomes periodic for any real number field
K as far as s + 1 = degQ (K) ≤ 4. But, it seems very likely that the algorithm
will not work well if degQ (K) = 5. In this paper we give a new algorithm and
discuss some properties and experimental results by which we can expect that the
expansion of α by the new algorithm always becomes periodic for any real number
field K with degQ (K) ≤ 5.
1. Introduction
Among problems related to higher dimensional continued fractions, a central problem has been to find a higher-dimensional generalization of Lagrange’s theorem
concerning the periodic continued fractions. Related to this problem the following
conjecture has been believed.
Conjecture. Let 1, α1 , . . . , αs be a Q−basis of real number field K with [K :
Q] = s + 1. Then, the expansion of (α1 , . . . , αs ) by the Jacobi-Perron algorithm is
eventually periodic, cf. [7].
But this conjecture may be false even in the case where s = 2, cf. [8]. We also
presented an algorithm in [8](say, the algebraic Jacobi-Perron Algorithm) which
is something like the Jacobi-Perron algorithm and discussed some experiments by
which we can expect that the expansion of α = (α1 , . . . , αs ) with some natural
conditions by the algorithm always becomes periodic for any real field K with
s + 1 = degQ (K) ≤ 4.
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INTEGERS: 11B (2011)
In this paper, we have two main objectives:
(1) To give computer experiments of expansions of α = (α1 , . . . , αs ) ∈ K s (with
some natural conditions on α) obtained by the algorithm [8] by which we can
expect that the expansions are not always periodic, cf. Tables E.
(2) To introduce a new algorithm and give experimental results according to the
new algorithm by which we can expect that the resulting expansion of α always
becomes periodic for any α ∈ K s (with some natural conditions on α) and any
real number field K with degQ (K) = s + 1 ≤ 5, cf. Tables A-E. We also give
experiments for α ∈ K 5 with degQ (K) = 6 such that the experiments for some
α obtained by our new algorithm are expected to be non periodic except for
some accidental cases. (cf. Tables F, and a problem given in Section 6.)
2. Cubic Case
In this section, we shall give two algorithms AJPA1 and AJPA2. Throughout this
section K denotes a real cubic field (including totally real cases and not totally real
cases). First, we define Algebraic Jacobi-Perron Algorithm(AJPA) presented in [8],
which will be referred to as AJPA1. We denote by XK the set defined by
XK := {(α, β) ∈ K 2 |1, α, β are linearly independent over Q} ∩ [0, 1)2 .
(1)
We define the transformation TK on XK by:
 %
& '
& '(
1
1
β
β


−
, −
if √ α
>√ β ,

|N (α)|
|N (β)|
(1)
α
α
α
α
%
&
'
&
'(
TK (α, β) :=
α
α
1
1


−
, −
if √ α
<√ β

|N (α)|
|N (β)|
β
β β
β
for (α, β) ∈ XK , where &x' is the floor function of x and N (x) is the norm of x ∈ K
over Q.
We define the integer-valued functions a(1) , b(1) and e(1) for (α, β) ∈ XK by:
) 1 *
 α
a(1) (α, β) := + α ,

β
+ ,
β

 α
(1)
+
,
b (α, β) :=

 β1

0
e(1) (α, β) :=
1
β
,
|N (β)|
β
√
,
|N (β)|
if √
>√
if
<
α
|N (α)|
√ α
|N (α)|
β
,
|N (β)|
√ β ,
|N (β)|
if √
>√
if
<
α
|N (α)|
√ α
|N (α)|
β
,
|N (β)|
β
√
.
|N (β)|
if √
>√
if
<
α
|N (α)|
√ α
|N (α)|
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INTEGERS: 11B (2011)
We put (for n ∈ Z>0 ):
(1) (1)
(1)
(1)
(1)
(a(1)
n , bn , en ) = (an (α, β), bn (α, β), en (α, β))
(1)n−1
:= (a(1) (TK
S
(1)
(α, β) :=
(1)n−1
(α, β)), b(1) (TK
(1)n−1
(α, β)), e(1) (TK
(α, β)),
(1)
(1)
∞
{(a(1)
n (α, β), bn (α, β), en (α, β))}n=1 .
(1)
The sequence S (1) (α, β) will be referred to as the expansion of (α, β) ∈ XK by TK ;
(1)
TK gives rise to a 2-dimensional continued fraction expansion algorithm, which
will be called AJPA1.
Secondly, we define an algorithm, which will be referred to as AJPA2. For an
algebraic number θ, we mean by φθ ∈ Q[x]
minimal polynomial of θ. We
- d the monic
.
|θ|
√
define ν(θ) :=
, where D(θ) := dx φθ (x) x=θ is the differential coefficient of
|D(θ)|
φθ (x) at x = θ.
(2)
We define the transformation TK on XK by:
 %
& '
& '(
1
1
β
β


−
, −
if



α
α
α
α


(2)
%
& '
& '( or
TK (α, β) :=
α
α
1
1


−
, −
if



β
β
β
β


or
ν(α) > ν(β) ,
ν(α) = ν(β) with α > β,
ν(α) < ν(β) ,
ν(α) = ν(β) with α < β.
We define the integer valued functions a(2) , b(2) and e(2) on XK as follows:
& '
1


if ν(α) > ν(β) or ν(α) = ν(β) with α > β,

α
a(2) (α, β) := & '
α


if ν(α) < ν(β) or ν(α) = ν(β) with α < β,

β
& '
β


if ν(α) > ν(β) or ν(α) = ν(β) with α > β,

α
(2)
b (α, β) := & '
1


if ν(α) < ν(β) or ν(α) = ν(β) with α < β,

β
/
0
if ν(α) > ν(β) or ν(α) = ν(β) with α > β,
(2)
e (α, β) :=
1
if ν(α) < ν(β) or ν(α) = ν(β) with α < β
for (α, β) ∈ XK .
We put (for n ∈ Z>0 )
(2) (2)
(2)
(2)
(2)
(a(2)
n , bn , en ) = (an (α, β), bn (α, β), en (α, β))
(2)n−1
:= (a(2) (TK
S
(2)
(α, β) :=
(2)n−1
(α, β), b(2) (TK
(2)n−1
(α, β)), e(2) (TK
(2)
(2)
∞
{(a(2)
n (α, β), bn (α, β), en (α, β))}n=1 .
(α, β)),
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INTEGERS: 11B (2011)
(2)
The sequence S (2) (α, β) will be referred to as the expansion of (α, β) ∈ XK by TK ;
TK gives rise to a 2-dimensional continued fraction expansion, which will be called
as AJPA2. Throughout our paper αn , βn are numbers defined by
(2)n
(αn , βn ) := TK
(2)
(2)
(α, β), (n ∈ Z≥0 ).
(2)
Notice that (an , bn , en ) ∈ Z≥0 × Z≥0 × {0, 1}, (n ∈ Z≥0 ).
For each (a, b, e) ∈ Z>0 × Z≥0 × {0, 1}, we put


0 0 1







0 1 b  if e = 0,





 1 0 a
A(a,b,e) :=
(1)





1 0 a







0 0 1 if e = 1,




0 1 b
 $$

pn (α, β) p$n (α, β) pn (α, β)
Mn (α, β) = qn$$ (α, β) qn$ (α, β) qn (α, β) := A(a(2) ,b(2) ,e(2) ) · · · A(a(2) ,b(2) ,e(2) ) .
n
n
n
1
1
1
rn$$ (α, β) rn$ (α, β) rn (α, β)
(2)
AJP A1
AJP A2
Definition (The set of periodic points.) We define PK
and PK
as follows:
AJP A1
PK
= {(α, β) ∈ XK | there exist m, n ∈ Z>0 such that m )= n and
(1)m
TK
AJP A2
PK
(1)n
(α, β) = TK
(α, β)},
= {(α, β) ∈ XK | there exist m, n ∈ Z>0 such that m )= n and
(2)m
TK
(2)n
(α, β) = TK
(α, β)}.
(j)
AJP Aj
If (α, β) ∈ PK
, the expansion S (j) (α, β) by TK becomes periodic, and vice
versa for each j = 1, 2. In what follows we mean by ”the period” the period obtained
by choosing the shortest period and preperiod. For the periodic continued fraction
obtained by AJPA2, we have the following Proposition 1 ,which can be shown in a
way similar to Perron [5].
AJP A2
Proposition 1. Let (α, β) ∈ PK
. Then, there exists a constant c(α, β) > 0
and η(α, β) > 0 such that η(α, β) ≤ 32 and both
|α −
pn
c(α, β)
| ≤ η(α,β)
rn
rn
hold. Further, η(α, β) =
3
2
and
|β −
qn
c(α, β)
| ≤ η(α,β)
rn
rn
holds if and only if K is not a totally real cubic field.
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INTEGERS: 11B (2011)
Remarks.
1. Based on our many experiments ( cf. Tables A, B, C, and D), we can hope
AJP A2
that XK = PK
.
2. In view of Proposition 1, we see that (α, β) )= (α$ , β $ ) if and only if S (2) (α, β) )=
AJP A2
S (2) (α$ , β $ ), for (α, β), (α$ , β $ ) ∈ PK
.
Bernstein [1] gave some classes of periodic continued fractions obtained by the
Jacobi-Perron algorithm. We can also give some examples of periodic expansions
obtained by AJPA2 in the similar manner as in [8].
√
√
Theorem
2. Let K = Q( 3 m3 + 1) with m ∈ Z>0 . Let (α, β) = ( 3 m3 + 1 −
6
AJP A2
m, 3 (m3 + 1)2 − m2 ). Then, (α, β) ∈ PK
.
√
Proof. Let m ≥ 4. We put u = 3 m3 + 1. We get S (2) (α, β) as in the following
tables. We prove the cases where n = 1 in the tables. We can prove the other cases
in the similar manners.
We see easily that |D(α)| = 3u2 and |D(β)| = 3u(m3 + 1).
We have
%
(
β2
α2
|D(α)D(β)|
−
(3)
|D(β)| |D(α)|
= (−2m2 u2 + (m3 + 1)u + m4 )(3u2 ) − 3u(1 + m3 )(u2 − 2mu + m2 ).
Putting u = m + ', we have
%
(
β2
α2
|D(α)D(β)|
−
|D(β)| |D(α)|
Since
= −9'm5 − 27'2 m4 − 24'3 m3 + 3y 3 − 6'4 m2 + 9'm2 + 6'2 m.
(m + ')3 = m3 + 1,
(4)
we have 3m3 = 9'm5 + 9'2 m4 + 3'3 m3 and 9'm2 = 27'2 m4 + 27'3 m3 + 9'4 m2 ,
which implies
− 9'm5 − 27'2 m4 − 24'3 m3 + 3m3 − 6'4 m2 + 9'm2 + 6'2 m
= −18'2 m4 − 21'3 m3 + 9'2 m3 − 6'4 m2 + 9'm2 + 6'2 m
= 9'2 m4 + 6'3 m3 + 9'2 m3 + 3'4 m2 + 6'2 m > 0.
Thus, we have
β2
α2
|D(β)| − |D(α)| >
1
u+m < 1, which
(2)
0. Therefore, we have e1 = 1. Then, we see that
= uu−m
implies a1 = 0. We have
2 −m2 =
We are going to prove the following inequalities:
α0
β0
1
β0
=
3m 1
m2 u2 + (m3 + 1)u + m4
3m
+ >
>
.
2
2
2m3 + 1
2
m2 u2 +(m3 +1)u+m4
.
2y 3 +1
(5)
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INTEGERS: 11B (2011)
First, we see that
m2 u2 + (m3 + 1)u + m4
3m
6'm3 + 2'2 m2 − m + 2'
−
=
.
2m3 + 1
2
4m3 + 2
On the other hand, by virtue of 3'm2 + 3'2 m + '3 = 1 we see easily that
1
6%m3 +2%2 m2 −m+2%
3m2 . Hence, we get
4m3 +2
3m
2 . Secondly, by (5) we see that
> 0. Therefore, we have
1
6m2 < '
m u +(m +1)u+m4
2m3 +1
2
2
3
<
>
3m 1 m2 u2 + (m3 + 1)u + m4
−6'm3 + 2m3 − 2'2 m2 + m − 2' + 1
+ −
=
2
2
2m3 + 1
4m3 + 2
3
(−3' + 1)m + (1 − 2'2 m)m + (−2' + 1)
=
4m3 + 2
> 0.
(2)
Hence we get that b1 =
3m
2
(2)
for even m, and b1 =
3m
2
−
1
2
for odd m.
Case 1. m = 1.
n
(2)
an
(2)
bn
(2)
en
(2)
(2)
(2)
(2)
1
0
1
1
2
2
1
0
(2)
(2)
For n ≥ 3, an = an−2 , bn = bn−2 and en = en−2 .
n
0
1
αn
u−1
1+u2 −u
3
βn
u2 − 1
−2+u2 +2u
3
For n ≥ 2, αn = αn−2 and βn = βn−2 .
Case 2. m = 3.
n
(2)
an
(2)
bn
(2)
en
n
(2)
an
(2)
bn
(2)
en
1
0
4
1
15
2
5
1
2 3
6 0
3 5
0 1
16
6
0
0
(2)
4
5
2
0
17
3
4
1
(2)
5
0
6
1
18
9
0
0
(2)
6
4
3
0
19
0
3
1
7
0
9
1
8
3
0
0
20
0
27
1
(2)
9
27
0
0
21
0
3
1
10
3
0
0
22
9
0
0
(2)
11
0
9
1
(2)
For n ≥ 23, an = an−22 , bn = bn−22 and en = en−22 .
12
4
0
0
13
3
6
1
14
5
0
0
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INTEGERS: 11B (2011)
n
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
αn
−3 + u
9
1 2
3
55 + 55 u − 55 u
−3 + u
1
1 2
1
9 + 36 u − 18 u
−3 + u
1
1 2
1
29 + 29 u − 29 u
−3 + u
1 2
28 u
−3 + u
−18 + u2 + 3u
−123
27 2
41
61 + 244 u + 122 u
−9 + u2
−139
9 2
28
55 + 55 u + 55 u
2
−6 + u − u
−29
7 2
11
9 + 36 u + 18 u
2
−3 + u − 2u
−109
7 2
22
29 + 29 u + 29 u
2
u − 3u
9 2
−6 + 28
u +u
2
9 + u − 6u
−3 + u
9
1
3
2
61 + 244 u − 122 u
βn
−9 + u2
9 2
28
− 139
55 + 55 u + 55 u
2
−6 + u − u
7 2
11
− 29
9 + 36 u + 18 u
2
−3 + u − 2u
−109
7 2
22
29 + 29 u + 29 u
2
u − 3u
9 2
−6 + 28
u +u
2
9 + u − 6u
−3 + u
9
1
3
2
61 + 244 u − 122 u
−3 + u
9
1 2
3
55 + 55 u − 55 u
−3 + u
1
1 2
1
9 + 36 u − 18 u
−3 + u
1
1 2
1
29 + 29 u − 29 u
−3 + u
1 2
28 u
−3 + u
−18 + u2 + 3u
−123
27 2
41
61 + 244 u + 122 u
For n ≥ 22, αn = αn−22 and βn = βn−22 .
Case 3. m is even.
n
1
2
3
(2)
an
0
3m
2
1
11
0
2m
0
0
2m
0
12
2m
bn
2m
0
(2)
en
1
0
1
13
0
3m
2
1
(2)
bn
(2)
en
n
(2)
an
(2)
(2)
(2)
4
3m
2
0
5
6
7
8
9
0
m
3m2
m
0
3m
0
0
0
3m
10
3m
2
0
0
14
3m
1
15
0
0
16
0
1
0
0
m
3m2
m
0
0
1
1
1
0
(2)
(2)
0
17
0
(2)
0
18
3m
(2)
For n ≥ 19, an = an−18 , bn = bn−18 and en = en−18 .
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INTEGERS: 11B (2011)
n
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
αn
u−m
u2 − mu + m2
2m3 + 1
βn
u2 − m2
2m2 u2 − (−2m3 − 2)u − 4m4 − 3m
4m3 + 2
2u2 − mu − m2
2
m2 u2 + (6m3 + 8)u − 12m4 − 12m
9m3 + 8
u2 − mu
m2 u2 + (m3 + 1)u − 2m4 − 2m
m3 + 1
u2 − 2um + m2
u−m
u2 − 2mu + 4m2
9m3 + 1
u−m
u2 − mu + m2
2m3 + 1
u−m
8u2 − 4mu + 2m2
9m3 + 8
u−m
u2
3
m +1
u−m
u2 + mu − 2m2
3m2 u2 + (3m3 + 1)u − 6m4 − 2m
9m3 + 1
u2 − m2
2m2 u2 − (−2m3 − 2)u − 4m4 − 3m
4m3 + 2
2u2 − mu − m2
2
m2 u2 + (6m3 + 8)u − 12m4 − 12m
9m3 + 8
u2 − mu
m2 u2 + (m3 + 1)u − 2m4 − 2m
m3 + 1
u2 − 2um + m2
u−m
u2 − 2mu + 4m2
9m3 + 1
u−m
8u2 − 4mu + 2m2
9m3 + 8
u−m
u2
m3 + 1
u−m
u2 + mu − 2m2
3m2 u2 + (3m3 + 1)u − 6m4 − 2m
9m3 + 1
For n ≥ 18, αn = αn−18 and βn = βn−18 .
Case 4. m is odd.
n
1
2
3
4
0
1
(3m
2
m
2m − 1
1
(m
2
1
0
1
n
10
11
(2)
an
1
(3m
2
(2)
an
0
(2)
bn
1
(3m
2
(2)
en
2m
− 1)
− 1)
m
5
2
8
9
m+1
3m
m
0
3m − 3
3
3
0
3m
0
1
0
0
0
1
12
13
14
15
16
17
18
2m − 1
1
(m
2
3m − 3
3
3
0
3m
0
m+1
3m
m
0
0
1
1
1
0
+ 1)
+ 1)
0
2m
0
1
(3m
2
(2)
en
0
1
0
1
(2)
7
0
(2)
bn
(2)
(2)
(2)
+ 1)
6
+ 1)
(2)
(2)
For n ≥ 19, an = an−18 , bn = bn−18 and en = en−18 .
2
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INTEGERS: 11B (2011)
n
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
αn
u−m
u2 − um + m2
2m3 + 1
u−m
8u2 + (−4m − 4)u + 2m2 + 4m + 2
9m3 + 3m2 + 3m + 9
u−m
u2 − u + 1
m3 + 2
u−m
u2 + mu − 2m2
3m2 u2 + (3m3 + 1)u − 6m4 − 2m
9m3 + 1
u2 − m2
2m2 u2 + (2m3 + 2)u − 4m4 + 2m3
4m3 + 2
−3m + 1
+
4m3 + 2
2u2 + (1 − m)u − m2 − m
2
(6m2 + 2)u2 + (6m3 + 2m + 8)u
9m3 + 3m2 + 3m + 9
−12m4 + 3m3 − m2 − 11m + 5
+
9m3 + 3m2 + 3m + 9
u2 + (1 − m)u − m
(m2 − m + 1)u2 + (m3 − m2 + m + 1)u
m3 + 2
−2m4 + 2m3 + m2 − 5m + 5
+
m3 + 2
u2 + (1 − m)u − m
u−m
u2 − 2mu + 4m2
9m3 + 1
βn
u2 − m2
2m2 u2 + (2m3 + 2)u − 4m4 + 2m3
4m3 + 2
−3m + 1
+
4m3 + 2
2u2 + (1 − m)u − m2 − m
2
(6m2 + 2)u2 + (6m3 + 2m + 8)u
9m3 + 3m2 + 3m + 9
−12m4 + 3m3 − m2 − 11m + 5
+
9m3 + 3m2 + 3m + 9
u2 + (1 − m)u − m
(m2 − m + 1)u2 + (m3 − m2 + m + 1)u
m3 + 2
−2m4 + 2m3 + m2 − 5m + 5
+
m3 + 2
m2 + (−2u − 3)m + u2 + 3u
u−m
u2 − 2mu + 4m2
9m3 + 1
u−m
u2 − um + m2
2m3 + 1
u−m
8u2 + (−4m − 4)u + 2m2 + 4m + 2
9m3 + 3m2 + 3m + 9
u−m
u2 − u + 1
m3 + 2
u−m
u2 + mu − 2m2
3m2 u2 + (3m3 + 1)u − 6m4 − 2m
9m3 + 1
For n ≥ 18, αn = αn−18 and βn = βn−18 .
Theorem 3. Let δm be the root of x3 − mx + 1 = 0 (m ∈ Z, m ≥ 3) determined by
2
0 < δm < 1. Then, K = Q(δm ) is a totally real cubic number field and (δm , δm
)∈
AJP A2
PK
.
Proof. We see easily that K = Q(δm ) is a totally real cubic number field. We
2
put (α, β) = (δm , δm
). We get S (2) (α, β) as in the following tables. We prove
2
the cases where n = 1 in the tables. We see easily that |D(α)| = m − 3δm
and
2
2
|D(β)| = m − mδm − 3δm .
10
INTEGERS: 11B (2011)
We have
|D(α)D(β)|
%
α2
β2
−
|D(α)| |D(β)|
(
(6)
2
2
2
2
= δm
(m2 − mδm
− 3δm ) − (m − 3δm
)(mδm
− δm )
=
4
2δm
m
+ δm m −
(7)
3
6δm
.
3
On the other hand, by δm
− mδm + 1 = 0 we have
δm =
which implies
1
m
2
m.
< δm <
3
δm
+1
,
m
(8)
Therefore, we get
4
3
3
3
2δm
m + δm m − 6δm
> 2δm
+ δm m − 6δm
3
= δm m − 4δm
> δm (m −
24
)
m2
> 0.
Thus, we have e1 = 0. Then, we see that αβ00 = δm , which implies b1 = 0. We have
1
2
2
β0 = m − δm , which implies a1 = m − 1. Hence, we have α1 = 1 − δm and β1 = δm .
(2)
(2)
(2)
n
1
2
an
m−1
1
bn
0
0
en
0
0
3
0
1
4
0
m−1
1
1
5
m−2
1
0
1
0
0
1
m−2
1
6
7
(2)
(2)
(2)
αn−1
δm
2
1 − δm
2
−(m − 1)δm
− δm + 1
m(m − 2)
δm
−δm + 1
m−2
2
−δm
− δm + 1
2
−δm
+1
m−2
(2)
(2)
βn−1
2
δm
δm
2
−δm
− (m − 1)δm + m − 1
m(m − 2)
2
−δm
− δm + 1
2
−δm
+1
m−2
δm
−δm + 1
m−2
(2)
For n ≥ 8, an = an−4 , bn = bn−4 , en = en−4 , αn−1 = αn−5 and βn−1 = βn−5 .
3. General Definition
In Section 2, we defined AJPA2 for cubic cases, which can be generalized to any
real number field K with d = degQ (K) ≥ 3. In this section, we mean by XK and
11
INTEGERS: 11B (2011)
$
XK
the set defined by
XK := {(α1 , . . . , αd−1 ) ∈ (K ∩ I)d−1 | there exists a integer i with 1 ≤ i ≤ d − 1
such that K = Q(αi ) and 1, α1 , . . . , αd−1 are linearly independent over Q}.
The function ν(θ) defined in Section 2 can be extended to θ ∈ K by

θ

if K = Q(θ),
1
d−1
|D(θ)|
ν(θ) :=
−1
if K )= Q(θ),
(9)
-d
.
where dx
φθ (x) x=θ which is the differential coefficient of φθ (x) at x = θ as in
Section 2.
For α = (α1 , . . . , αd−1 ) ∈ XK , we define ρ(α) = max{ν(αi )|1 ≤ i ≤ d − 1}. We
denote by ω(α) the number i ∈ {1, . . . , d − 1} uniquely determined by
αi = max{αj |ρ(α) = ν(αj )}.
(2)
We define a transformation TK on XK as follows:
For α = (α1 , . . . , αd−1 ) ∈ XK , TK (α) = (β1 , . . . , βd−1 ) with
&
'

1
1


−
if i = ω(α),

αω(α) & αω(α) '
βi :=
αi
αi


−
if i )= ω(α)
(i = 1, . . . , d − 1).

αω(α)
αω(α)
(2)
(2)
We easily see that TK is well-defined on XK . The transformation TK gives rise
to an algorithm of continued fraction of dimension d − 1, which will be also referred
AJP A2
to AJPA2. We define PK
, in a similar fashion to Section 2, that is,
AJP A2
PK
: = {α ∈ XK |there exist m1 , m2 ∈ Z>0 such that m1 )= m2 and
(2)m1
TK
(2)m2
(α) = TK
(α)}.
We put, for pq (p, q ∈ Z are coprime)
p
dh( ) := max{&log10 |p| + 1', &log10 |q| + 1'},
q
The function dh can be extended to Q[x]: for g(x) =
n
7
i=0
dh(g) := max {dh(ai )}.
0≤i≤n
dh(0) := 0.
ai xi ∈ Q[x], put
12
INTEGERS: 11B (2011)
We define dh , dhAJPA2 and rdhAJPA2 , namely, for α = (α1 , . . . , αd−1 ) ∈ XK and
n ∈ Z≥0
dh(α) :=
max
d/deg(pi )
{dh(pi
i∈{1,...,d−1}
(2)n
dhAJPA2 (n; α) := dh(TK
rdhAJPA2 (n; α) :=
)},
(α)),
(2)n
dh(TK (α))
dh(α)
,
where pi = φαi ∈ Q[x] (i ∈ {1, . . . , d − 1}) is the monic minimal polynomial of αi .
We define dhAJPA2 and rdhAJPA2 for α = (α1 , . . . , αd−1 ) ∈ XK as follows:
dhAJPA2 (α) := sup dhAJPA2 (n; α),
n∈Z≥0
rdhAJPA2 (α) :=
dhAJPA2 (α)
.
dh(α)
The function dhAJPA2 (n; α) (resp., rdhAJPA2 (n; α)) is referred to as the nth decimal height of α (resp., the nth relative decimal height of α) with respect to the
AJPA2. The function dhAJPA2 (α) (resp., rdhAJPA2 (α)) is referred to as the decimal
height of α (resp.,the relative decimal height of α) with respect to the AJPA2.
For an algorithm A like as AJPA2 on XK , we can define dhA (n; α), rdhA (n; α),
A
dhA (α), rdhA (α) and PK
analogously.
4. Numerical Experiments
In this section, we compare the expansion obtained by AJPA1, AJPA2, and modified
Jacobi-Perron algorithm (abbr. MJPA). The modified Jacobi-Perron algorithm is a
classical one, which is defined as follows: For x = (x1 , . . . , xn ) ∈ [0, 1)n (1, x1 , . . . , xn
are linearly independent over Q), we define ψ(x) ∈ {1, . . . , n} and a transformation
T̄ by
xψ(x) = max{x1 , . . . , xn },
T̄ (x1 , . . . , xn ) = (u1 , . . . , un ),
where
 y
,
if i )= ψ(x),

x
ψ(x)
&
'
ui :=
1
1


−
, if i = ψ(x).
xψ(x)
xψ(x)
INTEGERS: 11B (2011)
13
This algorithm has connection with accelerated Brun’s algorithm [7],[3] and the
modified Jacobi-Perron algorithm by Podsypanin [6]. It is not difficult to see that
the expansion obtained by the modified Jacobi-Perron algorithm is eventually periodic if and only if it is eventually periodic by accelerated Brun’s algorithm.
We
√
√
3
(2)
3
2
computed the length of the periods of the expansion S (+ m,, + m ,) for all
√
m ∈ Z with 2 ≤ m ≤ 5000 ( 3 m ∈
/ Q) and these decimal heights, cf. Table
A. For the calculation of the tables, we used a √
computer equipped with GiNaC
√
3
AJP A2
[4] on GNU C++. We confirmed that (+ 3 m,, + m2 ,) ∈ PK
for all m with
√
2 ≤ m ≤ 5000( 3 m ∈
/ Q). There are no reports on the periodicity of such pairs numbers obtained by Jacobi-Perron algorithm or the modified Jacobi-Perron algorithms.
In [2] Elsner and Hasse gave numerical results for 36 pairs of cubic numbers with
respect to the Jacobi-Perron algorithm. Among the pairs, they found 14 periodic
cases, while the other 22 cases are suspicious to be periodic.
2
AJP A2
We confirmed that (+τm ,, +τm
,) ∈ PK
for all integers m with 3 ≤ m ≤ 5000,
3
where τm is the maximal root of x − mx + 1, cf. Table B. Notice that Q(τm ) is a
totally real cubic field for any integer m ≥ 3.
√
√
√
4
4
We computed the length of the period of the expansion of (+ 4 m,, + m2 ,, + m3 ,)
√
with m ∈
/ Q) obtained by AJPA2 and relative decimal heights for m all 2 ≤ m ≤
5000, cf. Table C.
√
√
√
√
4
4
AJP A2
We confirmed that (+ 4 m,, + m2 ,, + m3 ,) ∈ PK
for all 2 ≤ m ≤ 5000( m ∈
/
Q).
We computed
the length
of the period of the expansion of
√
√
√
√
√
5
5
5
(+ 5 m,, + m2 ,, + m3 ,, + m4 ,) with 5 m ∈
/ Q) obtained by TK for m all 2 ≤ m ≤
2000 and√relative√decimal√heights given above, cf. Table D. We confirmed that
√
√
5
5
5
AJP A2
5
(+ 5 m,, + m2 ,, + m3 ,, + m4 ,) ∈ PK
for
all 2 ≤
m ≤ 2000(
m∈
/ Q).
√
√
√
√
√
5
5
5
We compare the expansions of (+ 5 m,, + m2 ,, + m3 ,, + m4 ,) with 5 m ∈
/ Q)
obtained by AJPA1, AJPA2 and the modified Jacobi-Perron algorithm for 2 ≤
m ≤ 30, cf. Table E. We seek the periods of the cases related to each algorithms
in so far as the decimal heights are less than 20 and we terminate seeking the
periods if these are greater than 20 or equal to 20. We find all cases of periodicity
related to AJPA2, 8 cases of periodicity related to AJPA1 and no case of periodicity
related to√MJPA.√We also
try to √
seek the lengths
of √
the periods of the expansion of
√
√
√
6
6
6
6
6
6
(+ 6 m,, + m2 ,, + m3 ,, + m4 ,, + m5 ,) with m2 , m3 ∈
/ Q obtained by AJPA2,
AJPA1 and MJPA for 2 ≤ m ≤ 30 in so far as the decimal heights are less than 100
and we terminate seeking the periods if these are greater than 100 or equal to 100;
cf. Table F. We find 2 cases of periodicity related to AJPA2, 2 cases of periodicity
related to AJPA1 and no case of periodicity related to MJPA.
14
INTEGERS: 11B (2011)
5. A Conjecture
Conjecture
AJP A2
(1) Let K be a real number field with degQ (K) ≤ 5. Then, XK = PK
.
(2) Let K be a real number field with degQ (K) ≤ 5. Then, there exists an absolute
constant c independent of K and α ∈ XK such that rdhAJPA2 (α) ≤ c holds.
AJP A1
(3) For some real number fields K with degQ (K) = 5, XK )= PK
holds.
AJP A2
(4) For some real number fields K with degQ (K) = 6, XK )= PK
holds.
6. Problem
The algebraic quantity ν(θ) given by (9) in Section 3 plays an important role in the
definition of our Algorithm AJPA2. Is it possible to find a suitable modification of
A
the value ν(θ) such that the resulting algorithm A has as a property XK = PK
for
any real algebraic number field K with degQ (K) = 6?
7. Tables
In Table A, below, LA (m1 , m2 ), HA (m1 , m2 ) and RA (m1 , m2 ) are numbers defined
by
LA (m1 , m2 ) := the maximum value of the length of the shortest period of
√
√
√
3
the expansion of (+ 3 m,, + m2 ,) by AJPA2 for m1 ≤ m ≤ m2 with 3 m ∈
/ Q,
√
√
3
HA (m1 , m2 ) :=
max √
dhAJPA2 (+ 3 m,, + m2 ,),
m1 ≤m≤m2 , 3 m∈Q
/
√
√
3
3
RA (m1 , m2 ) :=
max√
rdh
(+
m,,
+
m2 ,),
AJPA2
3
m1 ≤m≤m2 ,
m∈Q,
/
which are well-defined by the periodicity.
15
INTEGERS: 11B (2011)
TABLE A
range of m
m1 ≤ m ≤ m2
2 ≤ m ≤ 200
201 ≤ m ≤ 400
401 ≤ m ≤ 600
601 ≤ m ≤ 800
801 ≤ m ≤ 1000
1001 ≤ m ≤ 1200
1201 ≤ m ≤ 1400
1401 ≤ m ≤ 1600
1601 ≤ m ≤ 1800
1801 ≤ m ≤ 2000
2001 ≤ m ≤ 2200
2201 ≤ m ≤ 2400
2401 ≤ m ≤ 2600
2601 ≤ m ≤ 2800
2801 ≤ m ≤ 3000
3001 ≤ m ≤ 3200
3201 ≤ m ≤ 3400
3401 ≤ m ≤ 3600
3601 ≤ m ≤ 3800
3801 ≤ m ≤ 4000
4001 ≤ m ≤ 4200
4201 ≤ m ≤ 4400
4401 ≤ m ≤ 4600
4601 ≤ m ≤ 4800
4801 ≤ m ≤ 5000
LA (m1 , m2 )
HA (m1 , m2 )
RA (m1 , m2 )
474
1062
1806
1586
3338
2910
3762
5642
8694
8586
5846
6958
8706
6966
5522
12790
7286
10818
8210
17014
14574
17818
16214
16402
11574
5
6
6
6
6
6
7
7
7
7
7
7
7
7
7
7
7
7
8
8
7
8
8
8
8
5/3
3/2
3/2
6/5
6/5
6/5
7/5
7/5
7/5
7/5
7/5
7/5
7/5
7/6
7/6
7/6
7/6
7/6
4/3
4/3
7/6
4/3
4/3
4/3
4/3
16
INTEGERS: 11B (2011)
TABLE B
range of m
m1 ≤ m ≤ m2
3 ≤ m ≤ 200
201 ≤ m ≤ 400
401 ≤ m ≤ 600
601 ≤ m ≤ 800
801 ≤ m ≤ 1000
1001 ≤ m ≤ 1200
1201 ≤ m ≤ 1400
1401 ≤ m ≤ 1600
1601 ≤ m ≤ 1800
1801 ≤ m ≤ 2000
2001 ≤ m ≤ 2200
2201 ≤ m ≤ 2400
2401 ≤ m ≤ 2600
2601 ≤ m ≤ 2800
2801 ≤ m ≤ 3000
3001 ≤ m ≤ 3200
3201 ≤ m ≤ 3400
3401 ≤ m ≤ 3600
3601 ≤ m ≤ 3800
3801 ≤ m ≤ 4000
4001 ≤ m ≤ 4200
4201 ≤ m ≤ 4400
4401 ≤ m ≤ 4600
4601 ≤ m ≤ 4800
4801 ≤ m ≤ 5000
LB (m1 , m2 )
HB (m1 , m2 )
RB (m1 , m2 )
1038
4588
5916
11352
13020
15666
18946
22396
30256
39788
28572
68866
54294
54198
56542
50394
69318
60490
81362
94904
102666
128206
119286
124616
109448
6
7
8
7
8
8
8
8
8
9
9
9
9
9
9
9
9
9
9
9
10
10
10
10
10
2
7/3
7/3
7/4
2
2
2
2
2
9/4
9/4
9/4
9/4
9/4
9/4
9/4
9/4
9/4
9/4
9/4
5/2
5/2
5/2
5/2
5/2
In Table B, LB (m1 , m2 ), HB (m1 , m2 ) and RB (m1 , m2 ) are numbers defined by
LB (m1 , m2 ) := the maximum value of the length of the shortest period of
2
the expansion of (+τm ,, +τm
,) by AJPA2 for m1 ≤ m ≤ m2 , where τm is a
maximal root of x3 − mx + 1,
HB (m1 , m2 ) :=
RB (m1 , m2 ) :=
max
2
dhAJPA2 (+τm ,, +τm
,),
max
2
rdhAJPA2 (+τm ,, +τm
,),
m1 ≤m≤m2
m1 ≤m≤m2
which are well-defined by the periodicity.
17
INTEGERS: 11B (2011)
TABLE C
range of m
m1 ≤ m ≤ m2
2 ≤ m ≤ 200
201 ≤ m ≤ 400
401 ≤ m ≤ 600
601 ≤ m ≤ 800
801 ≤ m ≤ 1000
1001 ≤ m ≤ 1200
1201 ≤ m ≤ 1400
1401 ≤ m ≤ 1600
1601 ≤ m ≤ 1800
1801 ≤ m ≤ 2000
2001 ≤ m ≤ 2200
2201 ≤ m ≤ 2400
2401 ≤ m ≤ 2600
2601 ≤ m ≤ 2800
2801 ≤ m ≤ 3000
3001 ≤ m ≤ 3200
3201 ≤ m ≤ 3400
3401 ≤ m ≤ 3600
3601 ≤ m ≤ 3800
3801 ≤ m ≤ 4000
4001 ≤ m ≤ 4200
4201 ≤ m ≤ 4400
4401 ≤ m ≤ 4600
4601 ≤ m ≤ 4800
4801 ≤ m ≤ 5000
LC (m1 , m2 )
HC (m1 , m2 )
RC (m1 , m2 )
2316
9822
14182
16770
12802
19116
27178
26578
39660
27726
32892
42564
57774
118830
41802
34758
72366
98418
74874
47918
99462
44928
43934
79794
78162
8
9
10
10
11
11
11
11
11
13
11
12
12
12
12
13
12
12
13
13
12
12
13
12
13
2
3/2
10/7
10/7
11/8
11/8
11/8
11/8
11/8
13/8
11/9
4/3
4/3
4/3
4/3
13/9
4/3
4/3
13/9
13/9
4/3
4/3
13/9
4/3
13/9
In Table C, LC (m1 , m2 ), HC (m1 , m2 ) and RC (m1 , m2 ) are numbers defined by
LC (m1 , m2 ) := the maximum value of the length of the shortest period of
√
√
√
4
4
the expansion of (+ 4 m,, + m2 ,, + m3 ,) by AJPA2 for m1 ≤ m ≤ m2
√
with m ∈
/ Q,
√
√
√
4
4
HC (m1 , m2 ) :=
max √
dhAJPA2 (+ 4 m,, + m2 ,, + m3 ,),
m1 ≤m≤m2 , m∈Q
/
√
√
√
4
4
RC (m1 , m2 ) :=
max √
rdhAJPA2 (+ 4 m,, + m2 ,, + m3 ,),
m1 ≤m≤m2 , m∈Q
/
which are well-defined by the periodicity.
18
INTEGERS: 11B (2011)
TABLE D
range of m
m1 ≤ m ≤ m2
2 ≤ m ≤ 200
201 ≤ m ≤ 400
401 ≤ m ≤ 600
601 ≤ m ≤ 800
801 ≤ m ≤ 1000
1001 ≤ m ≤ 1200
1201 ≤ m ≤ 1400
1401 ≤ m ≤ 1600
1601 ≤ m ≤ 1800
1801 ≤ m ≤ 2000
LD (m1 , m2 )
HD (m1 , m2 )
RD (m1 , m2 )
94548
235884
308576
406580
745932
897654
1176156
1073388
1000436
1528364
17
19
21
18
19
20
14
19
19
19
8/3
19/9
21/10
9/5
19/11
20/11
14/11
19/11
19/11
19/12
In Table D, LD (m1 , m2 ), HD (m1 , m2 ) and RD (m1 , m2 ) are numbers defined by
LD (m1 , m2 ) := the maximum value of the length of the shortest period of
√
√
√
√
5
5
5
the expansion of (+ 5 m,, + m2 ,, + m3 ,, + m4 ,) by AJPA2 for
√
m1 ≤ m ≤ m2 with 5 m ∈
/ Q,
√
√
√
√
5
5
2 ,, + 5 m3 ,, + 5 m4 ,),
HD (m1 , m2 ) :=
max √
dh
(+
m,,
+
m
AJPA2
m1 ≤m≤m2 , 5 m∈Q
/
√
√
√
√
5
5
2 ,, + 5 m3 ,, + 5 m4 ,),
RD (m1 , m2 ) :=
max √
rdh
(+
m,,
+
m
AJPA2
5
m1 ≤m≤m2 ,
m∈Q
/
which are well-defined by the periodicity.
19
INTEGERS: 11B (2011)
TABLE E
m
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
AJPA2
length of dhAJPA2
period
4
2
366
5
2584
8
2618
7
136
7
40
4
1194
6
1016
7
9536
8
42
6
1284
12
204
7
1748
9
2500
8
2058
7
3756
9
516
8
138
6
1950
11
18556
10
1160
8
2260
14
4884
9
216
7
2390
8
3906
9
450
7
126
8
9486
10
AJPA1
length of dhAJPA1
period
4
2
(4552)
(2654)
(5861)
(6359)
76
4
294
6
(2000)
(2686)
24
6
(2707)
90
(166)
(3772)
(4190)
(6745)
(4248)
(5175)
156
7
(2365)
(3946)
408
7
(2708)
(2445)
(4139)
(2271)
(1667)
(3853)
34
8
(4648)
MJPA
length of dhMJPA
period
(148)
(196)
(218)
(245)
(192)
(163)
(172)
(111)
(211)
(222)
(230)
(166)
(222)
(141)
(194)
(123)
(165)
(157)
(110)
(104)
(159)
(250)
(170)
(141)
(303)
(117)
(116)
(158)
(278)
√
√
√
√
√
5
5
5
For αm = (+ 5 m,, + m2 ,, + m3 ,, + m4 ,) with 1 < m ≤ 30 and 5 m ∈
/ Q,
we seek to find a period of the expansions related to each algorithms during the
decimal heights dh∗ (n; αm ) keeps being less than 20 and we terminate seeking to
find a period if it exceeds 20 or is equal to 20. The number n with parentheses in
the columns dh∗ denotes the minimum number n such that dh∗ (n; αm ) exceeds 20
or is equal to 20.
20
INTEGERS: 11B (2011)
TABLE F
m
2
3
5
6
7
10
11
12
13
14
15
17
18
19
20
21
22
23
24
26
28
29
30
AJPA2
length of dhAJPA2
period
5
3
(20935)
(17777)
(17543)
(18516)
(18080)
(17286)
(20031)
(16304)
(16774)
(17406)
(15241)
140
7
(15681)
(19786)
(15430)
(14375)
(19745)
(18464)
(16857)
(15433)
(15742)
(17328)
AJPA
length of dhAJPA1
period
8
3
(7427)
(6966)
(6601)
(8014)
(7158)
(6517)
(6836)
(6666)
(6607)
(6197)
(6955)
54
7
(7534)
(7184)
(6268)
(7043)
(7200)
(6490)
(6983)
(7385)
(6986)
(5962)
MJPA
length of dhMJPA
period
(1285)
(1226)
(1115)
(1047)
(1193)
(1055)
(1108)
(1023)
(1155)
(1183)
(1166)
(1014)
(1131)
(1268)
(1098)
(1030)
(1140)
(1106)
(1170)
(1126)
(1161)
(1260)
(1270)
√
√
√
√
√
√
6
6
6
6
6
For α$ m = (+ 6 m,, + m2 ,, + m3 ,, + m4 ,, + m5 ,) with 1 < m ≤ 30 and m2 ,
√
6
m3 ∈
/ Q), we seek to find a period of the expansions related to each algorithms
during the decimal heights dh∗ (n; α$ m ) keeps being less than 100 and we terminate
seeking the periods if it exceeds 100 or is equal to 100. The number n with parentheses in the columns dh∗ denotes the minimum number n such that dh∗ (n; αm )
exceeds 100 or is equal to 100.
References
[1] L. Bernstein; The Jacobi-Perron algorithm—Its theory and application. Lecture Notes in
Mathematics, Vol. 207 Springer-Verlag, Berlin-New York (1971).
[2] L. Elsner, H. Hasse; Numerische Ergebnisse zum Jacobischen Kettenbruchalgorithmus in reinkubischen Zahlkörpern. (German) Math. Nachr. 34 (1967), 95-97.
INTEGERS: 11B (2011)
21
[3] T. Fujita, S. Ito, and S. Ninomiya; Symbolic and geometrical characterizations of Kronecker
sequences by using the accelerated Brun algorithm, J. Math. Sci., Tokyo 7 No.2 (2000), 163193.
[4] GiNaC web site:http://www.ginac.de/.
[5] O. Perron; Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus,
Math.Ann.64 (1907), 1-76.
[6] E. V. Podsypanin; A generalization of continued fraction algorithm that is related to the
Viggo Brun algorithm (Russian), Studies in Number Theory (LOMI), 4., Zap. Naucn. Sem.
Leningrad. Otdel. Mat. Inst. Steklov, 67 (1977),184-194.
[7] F. Schweiger; Multidimensional continued fractions. Oxford Science Publications. Oxford University Press, Oxford, 2000.
[8] J. Tamura and S. Yasutomi; A new multidimensional continued fraction Algorithm, Math.
Comp. 78 (2009), 2209-2222.