Perfect squares arise in many fundamental questions in number theory.
The law is so surprising and definitive that mathematicians in the 18th century tried to generalize it to cover other cases, such as when the exponent is not 2 but 3 or even larger. Since that time, the search for a general notion of a reciprocity law has been a deep theme in number theory. Mathematicians cheered when the Local Langlands Correspondence, which had stubbornly resisted proof for more than 30 years, was finally proved in recent work by Michael Harris and Richard Taylor, and independently by Guy Henniart. This result represents a major milestone in algebraic number theory. The article appears in the January 2000 issue of Notices of the AMS .