Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl

PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
• # Artykuł - szczegóły

## Acta Arithmetica

2013 | 158 | 2 | 141-164

## Circles passing through five or more integer points

EN

### Abstrakty

EN
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 ≥ 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 ≤ R. There exists a polynomial w(x) such that
$W_mm ≥ (4^{m})/(m!) R^{2} w(log R)(1+o(1))$
where w(x) is an explicit polynomial of degree $2^{m-1}-1$.

141-164

wydano
2013

### Twórcy

• School of Mathematics, Cardiff University, 23 Senghennydd Road, Cardiff CF24 4AG, Wales, UK
• School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK