PROCEEDINGS of the
AMERICAN MATHEMATICAL SOCIETY
Volume 98. Number 2. October 1986
ON BINOMIAL UNITS OF CERTAIN CUBIC FIELD
KIYOTA OZEKI
Abstract.
Scarowsky has conjectured about a binomial unit in a cubic field. We
discuss a relation between binomial units and a diophantine equation.
In a recent paper, Scarowsky [3] has conjectured about a binomial unit in a cubic
field. We prove the following
Theorem. Let K = Q(6) be a cubic field, where 63 + 12a0 - 12 = 0, a > 0. If
£ = 1 — ad is the fundamental unit ofK, it follows that all solutions ofx3 + Ylaxy2 —
12y3 = lare (x, y) = (1,0), (1, a).
Lemma 1 [1]. // bO + c, where b + 0, +1,
is a positive unit in Z[6]
where
03 - PO2 + Q6 — R = 0, then nopçwer > 1 of bd + c can be a binomial unit.
Lemma 2 [2]. Let e = +6 + c be a unit in Z[6], where 03 - P62 + Q0 - R = 0.
// 03 = 0 (mod p2), where p is a prime, then p ♦ c and e" = u + v6 is impossible for
n > 1.
The Diophantine
equation
equation x3 + Ylaxy2 — 12y3 = 1 can be written as a norm
NK/Q{x - 6y) = 1.
All solutions of the Diophantine equation are given by x - 8y = (1 - a0)",
n G Z.
Using Lemma 1 and 2, it is sufficient to prove that e~" cannot be a binomial unit
for n G JV. It is easy to calculate an inverse unit e"1, e"1 = (1 + 12a3) + aO + a262.
Put e-" = a„ + ß„6 + y„62, where a0 = 1, ß0 = 0, y0 = 0, a, = 1 + 12a3, ßx = a,
Yi = a2.
Lemma 3.
(an,ß,„yn)=(l,0,0)M"(a),
M(a)
=
'l + 12a3
a
,2
12a2
1
a
12la
0
1 I
Received by the editors October 11, 1985.
1980 Mathematics Subject Classification (1985 Revision). Primary 12A30.
í 1986 American Mathematical
Society
0002-9939/86 $1.00 + $.25 per page
215
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216
Proof.
KIYOTA OZEKI
Calculating e ("+1>,we have
= {(1 + 12a3) + a0 + a2e2){an
= {a2an + aßn + yn)02 +(aa„
+ ßn0 + yß2)
+ ß„)0 +(1 + 12a3)a„
+ 12a2ßn + 12ay„.
Comparing coefficients of 0, 62 and constant term, we obtain
a„ + l ={1 + 12a3)a„ + 12a2ß„ + 12ay„,
ßn + i = aan
+ ßn,
Y„+ i = a2a.n + aßn + y„,
or in matrix expression, (an + l,ß„ + l,yn + l) = (a„,ßn,y„)M(a).
Therefore,
(an + 1,ßn + 1,yn + l)=(l,0,0)M"+i(a).
Proof of the Theorem. Generally we have to calculate M "(a) explicitly, using,
for example, the Jordan canonical form of Af(a). However, in our case, it is enough
to check that a„, ßn, y„ =£0 for n > 1. All entries of the first column of M"(a) are
greater than 1 + 12a3 for n ^ 2, so we obtain an, ß„, yn > 0.
References
1. B. N. Delone
and D. K. Faddeev.
77îe theory of irrationalities
of the third degree. Transi. Math.
Monos., vol. 10, Amer. Math. Soc, Providence, R. I., 1964.
2. B. Gordon and S. P. Mohanty, On a theorem of Delaunay and some related results. Pacific J. Math. 68
(1977), 399-409.
3. M. Scarowsky,
On units of certain cubic fields and the diophantine equation, x3 + y3 + ;3 = 3, Proc.
Amer. Math. Soc. 91 (1984),351-356.
Department of Mathematics, The Faculty
350 Minf.-Machi, Utsunomiya, Japan
of General Education,
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Utsunomiya University,