BREMNER, ANDREW

Osaka City University Repository

Bremner, A.
Osaka J. Math.
16 (1979), 353-356
A DIOPHANTINE EQUATION ARISING FROM
TIGHT 4-DESIGNS
ANDREW BREMNER
(Received July 5, 1978)
Ito [1,2] and Enomoto, Ito, Noda [3] show that there exist only finitely
many tight 4-designs, by proving that such a design gives rise to a unique rational
integral solution of the diophantine equation
(2y>-3γ = xφt-2) (1)
and then invoking a result of Mordell [4] to say that this equation has only
finitely many solutions in integers x,y. A privately communicated conjecture
is that (1) has only the Obvious' solutions (±x> ±y)=(l, 1)> (3, 3), with the impli-
cation that the only tight 4-designs are the Witt designs. We show here that
this is indeed the case.
We are exclusively interested in integral points on the curve (1), which is
a lightly disguised elliptic curve; standard arguments show that the group of
rational points has one generator of infinite order which may be taken to be
(3,3).
Suppose now that x, y are integers satisfying (1). Then there is an integer
w with
3^-2 = w2
2y2-3 = wx. ^ '
Clearly x,w,y are odd. Following Cassels [5] we write (2), in virtue of the
identity w2— 3x2+2wxv/^3=(w+xV^:3)29 in the form
We now work in the algebraic number field Q(θ) where Θ2=V — 3. It is easy
to check that the ring of integers of Q(θ) has Z-basis j l , 0, ϋ^- , ^ i^-J, that
the class-number is 1, and that the group of units is generated by { — ω, ω+0}
_ I _ r\2 _
where ω= - is a cube root of unity. The relative norm to O(V — 3) of
354 A. BREMNER
the fundamental unit ε=ω+θ, is ω.
Further,
primes in Q(θ):
Ί Λ / Λ"
 - is prime in Z[ω], and splits into two first degree
Now the left hand side of (3) is the product of the two factors — — x^
conjugate over Q(V — 3), so by unique factorisation we deduce that
where η is a unit of Q(θ) with relative norm 1 - the possibilities for η are±£3ιw,
±ω£3m+1, ±ω263)W+2, for some integer m. By changing the sign of y if necessary,
we may thus assume that
= (ω£)' (l +i0-
where ι=0,l,2 and £=£3=i(ll-30-302+503).
Write (4) as
where λ is one of three possibilities,
λ,= 1+±
λ2=|
λ3= -
Δι Δι
We now choose to work 37-adically.
Since E6= — 1 mod 37, we have upon putting m= 6n+r, 0<r<5,
where ξ is an integer of £)(#) which by direct calculation satisfies ξ= —150—503
mod 37.
/9-I-/93Accordingly, we require that the coefficient of -^-— in \Er be congruent
DlOPHANTINE EQUATION ARISING FROM TlGHΓ 4-DESIGNS 355
to zero modulo 37: and this is clearly equivalent to the coefficient of θ3 being
zero modulo 37.
From the following table we deduce that \Er can only be λ2 or
(absorbing an E* into E6n for convenience) where \3E~l= —
Coefficient modulo 37 of 03 in λ^:-
\Er
λ2£
r
λ3£'
r=0
19
0
15
1
6
27
28
2
14
3
12
3
13
30
20
4
1
27
18
5
2
18
0
In the case that λ=λ2we have
( 5 )
One can treat this exponential equation in the manner of Skolem [6], but it is
preferable to argue directly. Suppose in (5) that wφO, and let the highest
power of 37 that divides n, be s.
Now (l+37ξ)n=W7nξ+372(ΐ)ξ2+
= l+37nξ mod 37S+2
= l+37n(-150-503) mod 37S+2.
So equating to zero the coefficient of θ3 on the right hand side of (5) we obtain
7)
 mo
d 37S+2
i.e. 0 = —35^.37 mod 375+2, contradiction.
Hence n=0 is the only possibility for a solution in (5), and it does indeed result
in(#,;y)=(3,-3).
The case \=\3E~
1
 is treated in precisely the same way, resulting in the
single solution (x,y)=(l9l).
We have thus shown that the only integer solutions of (1) are indeed given
by(±*,±y)=(l,l),(3,3).
EMMANUEL COLLEGE, CAMBRIDGE
References
[1] N. Ito: On tight 4-designs, Osaka J. Math. 12 (1975), 493-522.
356 A. BREMNER
[2] N. Ito: Corrections and supplement to "On tight ^-designs", Osaka J. Math. 15
(1978), 693-697.
[3] H. Enomoto, N. Ito, R. Noda: Tight 4-desιgns, Osaka J. Math. 16 (1979), 39-43.
[4] LJ. Mordell: Diophantine equations, Academic Press, 1969, p. 276.
[5] J.W.S. Cassels: Integral points on certain elliptic curves, Proc. London Math.
Soc. (3) 14A (1965), 55-57.
[6] Th. Skolem: Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen,
8de Skand. mat. Kongr. Forh. Stockholm, 1934.


A Diophantine equation arising from tight 4-designs

Bremner, Andrew

Osaka University Knowledge Archive

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A Diophantine equation arising from tight 4-designs

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