Cole published a number of important papers, including The Diurnal Variation of Barometric Pressure (1892).

[2][3][4] On October 31, 1903, Cole famously made a presentation to a meeting of the American Mathematical Society where he identified the factors of the Mersenne number 267 − 1, or M67.

During Cole's so-called "lecture", he approached the chalkboard and in complete silence proceeded to calculate the value of M67, with the result being 147,573,952,589,676,412,927.

Upon completing the multiplication and demonstrating that the result equaled M67, Cole returned to his seat, not having uttered a word during the hour-long presentation.

The American Mathematical Society's Cole Prizes in Algebra and Number Theory are named in his honor.