Any number (not itself a square number) may be multiplied by a square
that will give a product 1 less than another square. The given number
must not itself be a square, because a square multiplied by a square
produces a square, and no square plus 1 can be a square. My remarks
throughout must be understood to apply to whole numbers, because
fractional soldiers are not of much use in war.
Now, of all the numbers from 2 to 99 inclusive, 61 happens to be the
most awkward one to work, and the lowest possible answer to our puzzle
is that Harold's army consisted of 3,119,882,982,860,264,400 men. That
is, there would be 51,145,622,669,840,400 men (the square of
226,153,980) in each of the sixty-one squares. Add one man (Harold), and
they could then form one large square with 1,766,319,049 men on every
side. The general problem, of which this is a particular case, is known
as the "Pellian Equation"--apparently because Pell neither first
propounded the question nor first solved it! It was issued as a
challenge by Fermat to the English mathematicians of his day. It is
readily solved by the use of continued fractions.
Next to 61, the most difficult number under 100 is 97, where 97 ×
6,377,352² + 1 = a square.
The reason why I assumed that there must be something wrong with the
figures in the chronicle is that we can confidently say that Harold's
army did not contain over three trillion men! If this army (not to
mention the Normans) had had the whole surface of the earth (sea
included) on which to encamp, each man would have had slightly more than
a quarter of a square inch of space in which to move about! Put another
way: Allowing one square foot of standing-room per man, each small
square would have required all the space allowed by a globe three times
the diameter of the earth.