## 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).

### MSC:

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

Zbl 1042.11039
Full Text:

### References:

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