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# Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

## Circles passing through five or more integer points

### Tom 158 / 2013

Acta Arithmetica 158 (2013), 141-164 MSC: Primary 11P21; Secondary 11N37, 11M41. DOI: 10.4064/aa158-2-3

#### Streszczenie

We find an improvement to Huxley and Konyagin's current lower bound for the number of circles passing through five integer points. We conjecture that the improved lower bound is the asymptotic formula for the number of circles passing through five integer points. We generalise the result to circles passing through more than five integer points, giving the main theorem in terms of cyclic polygons with $m$ integer point vertices.

Theorem. Let $m \geq 4$ be a fixed integer. Let $W_m(R)$ be the number of cyclic polygons with $m$ integer point vertices centred in the unit square with radius $r \leq R$. There exists a polynomial $w(x)$ such that $W_mm \geq \frac{4^{m}}{m!}R^{2} w(\log R) (1+o(1))$ where $w(x)$ is an explicit polynomial of degree $2^{m-1}-1$.

#### Autorzy

• Shaunna M. Plunkett-LevinSchool of Mathematics
Cardiff University
Cardiff CF24 4AG, Wales, UK
and
School of Mathematics
University of Bristol
University Walk
Bristol BS8 1TW, UK
e-mail
e-mail

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