JBMO-SL 2006 Problem 5

Determine all pairs $(m,n)$ of natural numbers for which $m^2=nk+2$ where $k=\overline{n1}$ .

EDIT. It has been discovered the correct statement is with $k=\overline{1n}$ .

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.