INDETERMINATE EQUATIONS 
467 
4 
(/?) when G is positive and a square, say c 2 ; 
in this case Diophantus puts Ax 2 + c 1 = (mx + c) 2 , and obtains 
2 me 
± A 
■ 771" 
(y) When one solution is known, any number of other 
solutions can be found. This is stated in the Lemma to 
VI. 15. It would be true not only of the cases + Ax 2 + G = y 2 , 
but of the general case A x 2 + Bx + C — y 2 . Diophantus, how 
ever, only states it of the case Ax l — G = y 2 . 
His method of finding other (greater) values of x satisfy 
ing the equation when one {x 0 ) is known is as follows. If 
A x 2 — C = q 2 , he substitutes in the original equation (x 0 + x) 
for x and (q — kx) for y, where k is some integer. 
Then, since A(x 0 + x) 2 — G = (q — kx) 2 , while Ax 0 2 — C = q 2 , 
it follows by subtraction that 
2 x(Ax 0 + kq) = x 2 {k 2 — A), 
whence 
x — 2 [Ax q + kq) / {k 2 — A), 
2 {Axq + kq) 
k 2 —A 
and the new value of x is x 0 + 
Form Ax 2 — c 2 = y 2 . 
Diophantus says (VI. 14) that a rational solution of this 
case is only possible when A is the sum of two squares. 
[In fact, if x = p/q satisfies the equation, and Ax 2 — c 2 = k 2 , 
we have 
Ap 2 = c 2 q 2 + k 2 q 2 
Form Ax 2 + C = y 2 . - 
Diophantus proves in the Lemma to VI. 12 that this equa 
tion has an infinite number of solutions when A 4- G is a square, 
i.e. in the particular case where x = 1 is a solution. (He does 
not, however, always bear this in mind, for in III. 10 he 
regards the equation 52æ 2 +12 —y 2 as impossible though 
52 + 12 = 64 is a square, just as, in III. 11, 266æ 2 —10 = y 2 
is regarded as impossible.) 
Suppose that A + G = q 2 ; the equation is then solved by 
h h 2