On the Diophantine equation \(y^2=4q^n+4q+1\) (Q1071797)

This paper deals with the equation \(y^2=4q^n+4q+1\) in the unknowns \(n,y\in\mathbb N\) and \(q\) is a fixed prime power. It arises from coding theory as Calderbank's equation. The methods used are the reviewer's hypergeometric method and algebraic considerations, but the equation is not completely solved. However, by an alternative use of the hypergeometric method the authors have recently shown in a forthcoming paper that the only solutions are \((y,n)=(2q+1,2)\) or, when \(q=3\), \((y,n)=(5,1),\;(11,3)\).