OMM 1995 Problem 4

Find $26$ elements of $\{1, 2, 3, ... , 40\}$ such that the product of two of them is never a square. Show that one cannot find $27$ such elements.

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.