HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information

# Integral points on circles

Abstract : Sixty years ago the first named author gave an example \cite{sch} of a circle passing through an arbitrary number of integral points. Now we shall prove: {\it The number $N$ of integral points on the circle $(x-a)^2+(y-b)^2=r^2$ with radius $r=\frac{1}{n}\sqrt{m}$, where $m,n\in\mathbb Z$, $m,n>0$, $\gcd(m,n^2)$ squarefree and $a,b\in\mathbb Q$ does not exceed $r(m)/4$, where $r(m)$ is the number of representations of $m$ as the sum of two squares, unless $n|2$ and $n\cdot (a,b)\in\mathbb Z^2$; then $N\leq r(m)$}.
Keywords :
Document type :
Journal articles
Domain :

Cited literature [2 references]

https://hal.archives-ouvertes.fr/hal-01986718
Contributor : Srinivas Kotyada Connect in order to contact the contributor
Submitted on : Saturday, January 19, 2019 - 9:10:19 AM
Last modification on : Monday, March 28, 2022 - 8:14:08 AM

### File

41Article16.pdf
Files produced by the author(s)

### Citation

A Schinzel, M Skalba. Integral points on circles. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2019, Atelier Digit_Hum, pp.140 - 142. ⟨10.46298/hrj.2019.5116⟩. ⟨hal-01986718⟩

Record views