(related to Problem: The Sultan's Army)
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 $65,$ $(5 \times 13),$ can be expressed in two ways, $1,105,$ $(5 \times 13 \times 17)$, in four ways, $32,045,$ $(5 \times 13 \times 17 \times 29)$, 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 $25,$ $(5 \times 5),$ in two ways, but only in one; yet $125,$ $(5 \times 5 \times 5),$ can be given in two ways, and so can $625,$ $(5 \times 5 \times 5 \times 5);$ 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 $15,$ $(3 \times 5),$ will not work, nor will $135,$ $(3 \times 3 \times 3 \times 5);$ 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, $45,$ $(3 \times 3 \times 5,$ or $9 \times 5) = 36 + 9.$ 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 $2 \times 5 \times 13,$ 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 $5 \times 5 \times 13 \times 17 \times 29,$ 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: $(400, 15),$ $(399, 32),$ $(393, 76),$ $(392, 81),$ $(384, 113),$ $(375, 140),$ $(360, 175),$ $(356, 183),$ $(337, 216),$ $(329, 228),$ $(311, 252),$ $(265, 300).$ Square the two numbers in each pair, add them together, and their sum will in every case be $160,225.$
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