Proper partial geometries with Singer groups and pseudogeometric partial difference sets

A partial geometry admitting a Singer group G is equivalent to a partial difference set in G admitting a certain decomposition into cosets of line stabilizers. We develop methods for the classification of these objects, in particular, for the case of abelian Singer groups. As an application, we show that a proper partial geometry Π = pg ( s + 1 , t + 1 , 2 ) with an abelian Singer group G can only exist if t = 2 ( s + 2 ) and G is an elementary abelian 3-group of order ( s + 1 ) 3 or Π is the Van Lint–Schrijver partial geometry. As part of the proof, we show that the Diophantine equation ( 3 m − 1 ) / 2 = ( 2 r w − 1 ) / ( 2 r − 1 ) has no solutions in integers m , r ⩾ 1 , w ⩾ 2 , settling a case of Goormaghtighʹs equation.