The theory of numbers, by Robert D. Carmichael ...

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have this factor. For, consider an equation in (5) or in (6)
in which these two numbers occur; this equation contains a
third number, and it is readily seen that this third number is
divisible by t. Then from one of the equations containing the
fourth number it follows that this fourth number is divisible
by t. Now let us divide each equation of system (6) through
by t2; the resulting system is of the same form as (6). If
any two numbers in this resulting system have a common prime
factor ti, we may divide through by t12; and so on. Hence if
a pair of simultaneous equations (6) exists then there exists a
pair of equations of the same form in which no two of the numbers m, n, p, q have a common factor other than unity. Let
this system of equations be
2-+ql2 = 2m12, Pi2 -q12 = 2nl2. (7)
From the first equation in (7) it follows that pi and qi are
both even or both odd; and, since they are relatively prime,
it follows that they are both odd. Evidently pl>ql. Then
we may write
pi =ql + 2a,
where a is a positive integer. If we substitute this value of
pi in the first equation of (7), the result may readily be put in
the form
(q +a)2 +a2 =m12 (8)
Since qi and mi have no common prime factor it is easy to see
from this equation that a is prime to both qi and mi, and hence
that no two of the numbers qi +a, a, mi have a common factor.
Now we have seen that if a, b, c are positive integers no two
of which have a common prime factor, while
a2 +b2 = 2,
then there exist relatively prime integers r and s, r>s, such
that
c = r2 +2 a=2rS, b =r2-2
or
c = r2+s2, a=r2-2, b = 2rs.