{"docId":5116,"paperId":5116,"url":"https:\/\/hrj.episciences.org\/5116","doi":"10.46298\/hrj.2019.5116","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-01986718","repositoryVersion":1,"repositoryLink":"https:\/\/hal.science\/hal-01986718v1","dateSubmitted":"2019-01-23 14:21:42","dateAccepted":"2019-01-23 15:40:09","datePublished":"2019-01-23 15:40:18","titles":{"en":"Integral points on circles"},"authors":["Schinzel , A","Skalba , M"],"abstracts":{"en":"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":[["sums of two squares"],["Gaussian integers 2010 Mathematics Subject Classification 11D25"],["11D09"],"[ MATH ] Mathematics [math]","[ MATH.MATH-NT ] Mathematics [math]\/Number Theory [math.NT]"]}