Vol. 68, No. 2, 1977

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ISSN: 0030-8730
On a theorem of Delaunay and some related results

Basil Gordon and S. P. Mohanty

Vol. 68 (1977), No. 2, 399–409
Abstract

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 pαa,pβb, p being a prime, then 𝜖n = u + v𝜃 is impossible for n0 in the following cases:

  1. When 1 α β and p is odd,
  2. When 2 α β and p = 2,
  3. When β α < 2β and p is odd,
  4. 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.

Mathematical Subject Classification
Primary: 10B05, 10B05
Milestones
Received: 3 November 1975
Published: 1 February 1977
Authors
Basil Gordon
S. P. Mohanty