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

9'2

THEORY OF NUMBERS

This is obviously a special case of the more general theorem:
II. There are no integers p, q, a, all different from zero, such
that
p4 _ q4 = a2. (2)
The latter theorem is readily proved by means of theorem
III of ~ 46. For, if we assume an equation of the form (2),
we have
(p4 - q4)p2q2 = p2q2a2 (3)
But, obviously,
(2 p2q2)2 + (p4 -_ 4)2 (p4 +q4)2 (4)
Now, from (3) we see that the numerical right triangle determined by (4) has its area p2q2(p4 - q4) equal to the square number p2q2a2. But this is impossible. Hence no equation of the
form (2) exists.
EXERCISES
I. Show that the equation a4-+4/44= y2 iS impossible in integers a, 3, y, all
of which are different from zero.
2. Show that the system p2-q2=km2, p2+q2=kn2 is impossible in integers
p, q, k, m, n, all of which are different from zero.
3*. Show that neither of the equations n4 —4-4=t4 12 is possible in integers
m, n, t, all of which are different from zero.
4*. Prove that the area of a numerical right triangle is not twice a square
number.
5*. Prove that the equation +4-+n4-=a2 is not possible in integers m, it, a all
of which are different from zero.
6*. In the numerical right triangle a2+b2=c2, not more than one-of the numbers a, b, c is a square.
7. Prove that the equation x2a+y2 —Z2k: implies an equation of the form
m +n = 2 - 2tt
8. Find the general solution in integers of the equation x2+2-y2 =t2.
9. Find the general solution in integers of the equation X2 —y2=z4.
Io. Obtain solutions of each of the following Diophantine equations:
X3+ay3+-Z3= 2t3,
X3+2y3+3z3 = t,3
X4y4- 4z4 = t4?
x4+ v,4-+z4= 2t4.