10.46298/hrj.2019.5116 https://hrj.episciences.org/5116 Schinzel , A A Schinzel Skalba , M M Skalba Integral points on circles 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 $N\leq r(m)$}. episciences.org sums of two squares Gaussian integers 2010 Mathematics Subject Classification 11D25 11D09 [ MATH ] Mathematics [math] [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT] 2019-01-23 2019-01-23 2019-01-23 en journal article https://hal.science/hal-01986718v1 2804-7370 https://hrj.episciences.org/5116/pdf VoR application/pdf Hardy-Ramanujan Journal Researchers Students