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

OTHER TOPICS

89

Now from (9) and (io) and the relations pl=ql+2a, r>s,
we see that
pl=2rs+r2 -2 > 22+r2- 2 Sr2- S2 = U4 +v4
Hence u<pi. Also,
i2 _ 2 rw2 r+s<r2+s2.
Hence wl<pl. Since u and wl are both less than pi it follows
that p2 is less than pi. Hence, obviously, P2<p. Moreover,
it is clear that all the numbers p2, q2, m2, n2 are different from
zero.
From these results we have the following conclusion: If
we assume a system of the form (6) we are led to a new system
(I3) of the same form; and in the new system p2 is less than p.
Now if we start with (13) and carry out a similar argument
we shall be led to a new system
P32+q32 = 2m32, 2 q32 - 22,
with the relation p3 <p2; starting from this last system we shall
be led to a new one of the same form, with a similar relation of
inequality; and so on ad infinitum. But, since there is only
a finite number of positive integers less than the given positive
integer p this is impossible. We are thus led to a contradiction; whence we conclude at once to the truth of II and likewise of I.
By means of theorems I and II we may readily prove the
following theorem:
III. The area of a numerical right triangle is never a square
number.
Let the sides and hypotenuse of a numerical right triangle
be u, v, w, respectively. The area of this triangle is ~uv. If
we assume this to be a square number t2 we shall have the
following simultaneous Diophantine equations

U2 +V2 =W-2,2 UV =2t2.

(I4)