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

88 THEORY OF NUMBERS
Hence from (8) we see that we may write
qli+a=2rs, a=r2-S2 (9)
or
qi+a=r2-s2, a= 2rs. (IO)
In either case we have
pl2 - ql2 = (pl-ql) (pl +ql) = 2. 2 (ql +a) = 8rs(r2- S2).
If we substitute in the second equation of (7) and divide by 2
we have
4rs(r2-s2) = n2.
From this equation and the fact that r and s are relatively
prime it follows at once that r, s, r2-s2 are all square numbers;
say,
r==2, s=- 2, r 2 2=w2.
Now r-s and r+s can have no common factor other than
i or 2; hence from
w2 = (r2 - S2) = (r-s) (r +s) = (u2 - 2) (U +v2)
we see that either
U2+V2=2W12, U2-_2V22==W22 (II)
or
U2+2 2= 12, U2 -12W22.
And if it is the latter case which arises, then
W12 +W22 =2U2, W12 -W22 ==2V2.2 (12)
Hence, assuming equations of the form (6) we are led either to
equations (ii) or to equations (I2); that is, we are led to new
equations of the form with which we started. Let us write
the equations thus:
p22S+22= 2122, p22-q22= 2422; (13)
that is, system (13) is identical with that one of systems (ii),
(I2) which actually arises.