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Delaunay has proved that if
𝜖 = apϕ2 + bpϕ + c is a unit in the ring Z[𝜃], where 𝜃3 − P𝜃2 + Q𝜃 − R = 0, p is an
odd prime, ϕ = pt𝜃, t ≧ 0 and p ∤ a, then no power 𝜖m (m positive) can be a binorm,
i.e. 𝜖m = u + v𝜃 is impossible for m a positive integer. Hemer has pointed out
that in the above situation, 𝜖m = u + v𝜃 is also impossible for m a negative
integer.
In this paper the above result is extended as follows.
Theorem 1. If 𝜖 = a𝜃2 + b𝜃 + c is a unit in Z[𝜃], where 𝜃3 = d𝜃2 + e𝜃 + f and
pα∥a,pβ∥b, p being a prime, then 𝜖n = u + v𝜃 is impossible for n≠0 in the following
cases:
- When 1 ≦ α ≦ β and p is odd,
- When 2 ≦ α ≦ β and p = 2,
- When β ≦ α < 2β and p is odd,
- When β ≦ α < 2β − 1 and p = 2.
As an application of this and some other similar theorems, all integer solutions of
the equation y2 = x3 + 113 are determined.
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