The smallest primes of the form 4n + 1 are 5, 13, 17, 29, and 37, and
the smallest of the form 4n - 1 are 3, 7, 11, 19, and 23. Now, primes of
the first form can always be expressed as the sum of two squares, and in
only one way. Thus, 5 = 4 + 1; 13 = 9 + 4; 17 = 16 + 1; 29 = 25 + 4; 37
= 36 + 1. But primes of the second form can never be expressed as the
sum of two squares in any way whatever.
In order that a number may be expressed as the sum of two squares in
several different ways, it is necessary that it shall be a composite
number containing a certain number of primes of our first form. Thus, 5
or 13 alone can only be so expressed in one way; but 65, (5 × 13), can
be expressed in two ways, 1,105, (5 × 13 × 17), in four ways, 32,045, (5
× 13 × 17 × 29), in eight ways. We thus get double as many ways for
every new factor of this form that we introduce. Note, however, that I
say _new_ factor, for the _repetition_ of factors is subject to another
law. We cannot express 25, (5 × 5), in two ways, but only in one; yet
125, (5 × 5 × 5), can be given in two ways, and so can 625, (5 × 5 × 5 ×
5); while if we take in yet another 5 we can express the number as the
sum of two squares in three different ways.
If a prime of the second form gets into your composite number, then that
number cannot be the sum of two squares. Thus 15, (3 × 5), will not
work, nor will 135, (3 × 3 × 3 × 5); but if we take in an even number of
3's it will work, because these 3's will themselves form a square
number, but you will only get one solution. Thus, 45, (3 × 3 × 5, or 9 ×
5) = 36 + 9. Similarly, the factor 2 may always occur, or any power of
2, such as 4, 8, 16, 32; but its introduction or omission will never
affect the number of your solutions, except in such a case as 50, where
it doubles a square and therefore gives you the two answers, 49 + 1 and
25 + 25.
Now, directly a number is decomposed into its prime factors, it is
possible to tell at a glance whether or not it can be split into two
squares; and if it can be, the process of discovery in how many ways is
so simple that it can be done in the head without any effort. The number
I gave was 130. I at once saw that this was 2 × 5 × 13, and consequently
that, as 65 can be expressed in two ways (64 + 1 and 49 + 16), 130 can
also be expressed in two ways, the factor 2 not affecting the question.
The smallest number that can be expressed as the sum of two squares in
twelve different ways is 160,225, and this is therefore the smallest
army that would answer the Sultan's purpose. The number is composed of
the factors 5 × 5 × 13 × 17 × 29, each of which is of the required form.
If they were all different factors, there would be sixteen ways; but as
one of the factors is repeated, there are just twelve ways. Here are the
sides of the twelve pairs of squares: (400 and 15), (399 and 32), (393
and 76), (392 and 81), (384 and 113), (375 and 140), (360 and 175), (356
and 183), (337 and 216), (329 and 228), (311 and 252), (265 and 300).
Square the two numbers in each pair, add them together, and their sum
will in every case be 160,225.