On certain plane curves with many integral points. (English) Zbl 1029.11025

Summary: The authors define a sequence of polynomials \(P_d\in \mathbb{Z}[x,y]\), such that \(P_d\) is absolutely irreducible, of degree \(d\), has low height, and has at least \(d^2+ 2d+3\) integral solutions to \(P_d(x,y)= 0\). No other nontrivial family of polynomials of increasing degree with as many integral solutions in terms of their degree is apparently known. (A. Bremner noticed that the point \((x,y)= (9,25)\) is missing from Table 1, line \(d=3\), and more generally \((x,y)= (4d-3, (2d-1)^2)\) as a further point of \(P_d\) for \(d\) odd.)
See also the authors and D. Zagier, Acta Arith. 99, 85-96 (2001; Zbl 1042.11039).


11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14G05 Rational points


Zbl 1042.11039
Full Text: DOI Euclid EuDML


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