Mathematical Recreations; April 2000; Scientific American Magazine; by Stewart; 2 Page(s)
In his 1917 book Amusements in Mathematics, English puzzle maker Henry Ernest Dudeney described a fanciful problem based on the Battle of Hastings, the famous confrontation in 1066 between the Saxons under King Harold and the Normans under William the Conqueror. According to Dudeney, an ancient chronicle of the battle stated: "The men of Harold stood well together, as their wont was, and formed sixty and one squares, with a like number of men in every square thereof.... When Harold threw himself into the fray the Saxons were one mighty square of men." What, asked Dudeney, is the smallest possible number of men in King Harold's army?
Mathematically, we want to find a perfect square that, when multiplied by 61 and increased by 1, yields another perfect square. That is, we want integer solutions of the equation y2 = 61x2 + 1. This is an example of a Pell equation, mistakenly named after an obscure 17th-century English mathematician whose contributions to the field were not original. Equations of this general kind-in which 61 can be replaced by any nonsquare positive integer-always have infinitely many solutions. The technique for calculating the solutions is called the continued-fractions method, which can be found in most number theory textbooks.