When is the Product of Two Oblong Numbers Another Oblong?

For centuries, mathemiiaticianis have been fascinated by the association of nuilmbers with geom-letric patterns, such as squiares or triacngles. A less-studied exam-nple of this geilre obloncg numbers-is the occasion for a nice student investigation: find pairs of oblong numbers whose product is also an oblong number. This activity has hiistorical resonance, and it involves the glatherin-g of data and the search for patterns, the formiulation of general results, and the use of Pell's equation. Oblongo uiuinbers are numbers of the formii ca(a + 1), where a is a positive integer. The first few oblong numu-bers are 2, 6, 12, 20, and 30. The ncamiie refers to the geometric formn by wlhich these numiibers miiay be represented.