Andrew Bremner

On the equation

Generators are found for the group of rational points on the title curve for all primes p 5 (mod 8) less than 1,000. The rank is always I in accordance with conjectures of Selmer and Mordell. Some of the generators are rather large.

On arithmetic progressions on elliptic curves

We study arithmetic progressions in the x-coordinates of rational points on elliptic curves. An infinite family of elliptic curves is found, each containing an arithmetic progression of length 8.

Which integers are representable as the product of the sum of three integers with the sum of their reciprocals

For example, the integer 564 is so representable by the three integers 122 44200 5010002877 81163 51171 17995 2136135134 91867, -3460 69586 84255 04865 64589 22621 88752 08971 30654 24460 and 74807 19101 53025 27837 94583 60171 46464 94820 59055 28060.

On power bases in cyclotomic number fields

The current paper considers the question of power bases in the cyclotomic number field Q(ζ), ζp = 1, p an odd prime. The ring of integers is Z[ζ], and there do exist further “non-obvious” generators for this order; specifically we shall see that Z[α] = Z[ζ] for α = ζ + ζ2 + ⋯ + ζ(p−1)2. We conjecture that, up to conjugacy, there can be no further such integral generators, and prove that this is indeed the case in Q(ζ7).

Rational points in geometric progressions on certain hyperelliptic curves

We pose a simple Diophantine problem which may be expressed in the language of geometry. Let C be a hyperelliptic curve given by the equation y2 = f(x), where f ∈ Z[x] is without multiple roots. We say that points Pi = (xi, yi) ∈ C(Q) for i = 1, 2, . . . , k, are in geometric progression if the numbers xi for i = 1, 2, . . . , k, are in geometric progression. Let n ≥ 3 be a given integer. In this paper we show that there exist polynomials a, b ∈ Z[t] such that on the curve y2 = a(t)xn + b(t) (defined over the field Q(t)) we can find four points in geometric progression. In particular this result generalizes earlier results of Berczes and Ziegler concerning the existence of geometric progressions on Pell type quadrics y2 = ax2 + b. We also investigate for fixed b ∈ Z, when there can exist rationals yi, i = 1, ..., 4, with {y2 i − b} forming a geometric progression, with particular attention to the case b = 1. Finally, we show that there exist infinitely many parabolas y2 = ax + b which contain five points in geometric progression.

TRIANGLE-RECTANGLE PAIRS WITH A COMMON AREA AND A COMMON PERIMETER

We solve a problem of Bill Sands, to find pairs of Heron triangles and rectangles, such as (5,5,6) & [2 × 6] or (13,20,21) & [6 × 21] which have a common area and a common perimeter. The original question was posed for right-angled triangles, but there are no nondegenerate such. There are infinitely many isosceles triangles and these have been exhibited by Guy. Here we solve the general problem; the triangle-rectangle pairs are parametrized by a family of elliptic curves.

The delta-lambda configurations in tiling the square

Discussion des configurations delta-lambda dans la triangulation du carre. Caracterisation et generation de ces configurations

ON xa ± yb ± zc ± wd = 0, 1/a + 1/b + 1/c + 1/d = 1

It is well known that the Diophantine equations x4 + y4 = z4 + w4 and x4 + y4 + z4 = w4 each have infinitely many rational solutions. It is also known for the equation x6 + y6 - z6 = w2. We extend the investigation to equations xa ± yb = ±zc ± wd, a, b, c, d ∈ Z, with 1/a + 1/b + 1/c + 1/d = 1. We show, with one possible exception, that if there is a solution of the equation in the reals, then the equation has infinitely many solutions in the integers. Of particular interest is the equation x6 + y6 + z6 = w2 because of its classical nature; but there seem to be no references in the literature.

Cyclic sextic trinomials x6 + Ax + B

A correspondence is obtained between irreducible cyclic sextic trinomials x6 + Ax + B ∈ ℚ[x] and rational points on a genus two curve. This implies that up to scaling, x6 + 133x + 209 is the only cyclic sextic trinomial of the given type.

On the Equation

We show that there are infinitely many nonisomorphic curves Y 2 = X 5 + k, k ∈ ℤ, possessing at least twelve finite points for k > 0, and at least six finite points for k < 0. We also determine all rational points on the curve Y 2 = X 5 – 7.