On arithmetic progressions on elliptic curves. (English) Zbl 0951.11021

Bremner, Andrew

Exp. Math. 8, No. 4, 409-413 (1999).

This is a masterly combination of elementary mathematics and nontrivial symbolic computations (performed with MAPLE V and APECS) for discovering elliptic curves possessing rational “arithmetic progressions”, i.e. rational points \((x,y)\) whose \(x\)-coordinates are in arithmetic progression. As the author observes, this is an instance of a problem that interrelate two completely different group structures, hence should be expected to be a difficult one. A very concrete method is described for constructing elliptic curves over \(\mathbb{Q} (m/n)\) (\(m\) and \(n\) should be understood here as rational parameters), each of which possesses a rational arithmetic progression of length 7. Then the author discusses the possibility of one further step for extending the length to 8: one is led to the condition that a homogeneous polynomial in \(m,n\) be a square. This corresponds to finding a rational point on a curve of genus \(>1\), hence, in each case, at most finitely many such points can exist. It is not clear to the reviewer why at least one such point does exist, as the author seems to suggest.

Reviewer: Nikos Tzanakis (Iraklion)

MSC:

11G05	Elliptic curves over global fields
11B25	Arithmetic progressions

Keywords:

arithmetic progression on elliptic curve

Software:

APECS; Maple

References:

[1]	Bremner A., J. Number Theory 79 (1999)
[2]	Buhler J. P., Math. Comp. 44 (170) pp 473– (1985)
[3]	Connell I., ”APECS: Arithmetic of Plane Elliptic Curves”
[4]	Lee J.-B., Period. Math. Hungar. 25 pp 31– (1992) · Zbl 0757.11009 · doi:10.1007/BF02454382