I am in the process of trying to devise analytic geometric methods for digital geometry, and came across the following equation:
$[\sqrt{x^2+y^2}]=5$, where $[x]$ denotes conventional rounding.
Apart from (obviously) noting that $0\le x,y\le5$, I let Wolfram Alpha solve this for me and it quickly gave $28$ integer solutions. Turning to the general case, $[\sqrt{x^2+y^2}]=k$, I am wondering if there is a known alternative method to solving for all integer solutions, apart from checking all possible options based on an inequality. I acknowledge it is a quadratic/nonlinear Diophantine equation, but I have never investigated these much before. Thanks!