Infrastructure: Structure Inside the Class Group of a Real Quadratic Field
Michael J. Jacobson, Renate Scheidler · Notices of the American Mathematical Society · 2013
Suppose that you are a wise ancient Greek, and that you have been given the task of counting the Sun God's cattle on the island of Sicily.They are too numerous to count manually, and your only clue is a puzzle that relates the number of cows and bulls of a particular color to those of another.How can you determine the size of your herd from such cryptic information?A precise version of this numerical puzzle is the well-known cattle problem of Archimedes [2].A number of accounts of its solution appear in the literature [31,21,19].Unfortunately for the hypothetical Greek sage, this problem does not have a small and completely elementary solutionit would take more than 2,000 years before the first solution was discovered by Amthor [1].Had a solution been available, the Greek would likely have been shocked to find that the Sun God had somehow managed to pack a herd of more than 10 206544 animals onto Sicily!One of the main ingredients in Amthor's solution was solving the Diophantine equation x 2 -4729414y 2 = 1 for integers x and y.This equation is a specific instance of the (incorrectly named) Pell equation (0.1)x 2 -Dy 2 = 1 , where D is assumed to be a positive nonsquare