118 Mr Pocklington, Some Diophantine Impossibilities. 
12. Some equations of the type considered can be solved — 
completely. We choose as an example a‘— 4a°?+y'=2 and 
confine our attention to the case of x prime to y, as it is clear 
that when all such solutions are known the others can be 
immediately derived from them. We cannot have both # and — 
y odd. Suppose y to be even. Then writing the equation 
(e+ ye 6a°y? = 22 we have a?+y? odd, and either 
e+ ypP=u + be, «sy =2u0, 
with w odd and (using modulus 4) v even, or 
e+y=sur + 2%, sy = 2w, 
with u odd and (using modulus 4) v also odd. In the first case 
a+ y?= 1, mod. 8, in the second case a? + y?= 5, mod. 8. 
In the first case x= a8, y= 276, u=ary, v = BO, with a, B, y odd 
and 6 even. The other equation now is 6? (a? — 662) = 9 (a? — 46°), 
which gives a?—68=9?, a? — 46?= 6°, the alternatives being im- 
possible to modulus 8. From the second of these a=&+ 77, 
= &n, and the first becomes (£+ 77)? — 6&n?= 9. Also 
&n =8< 2aBys < ay. 
Of course & is prime to 7. 
In the second case = a8, y = 2rd, u= ay, v = BO, but a, B, x, 6 
are all odd. The other equation now is a? (8? — 3y°) = 28? (3? — 2ry?), 
which gives 3n?— PB’ = 26%, 2n?— 6’ =e’, the alternatives being 
impossible to modulus 8. These give 
(B +4) (B—a)=4(y +6) (y— 8). 
We suppose none of the factors to vanish. Each of the factors 
is even, one of those on the left is not divisible by 4, and the 
other must be divisible op 8. Supposing * it to be 8+a we have 
B+a ae y+6 yn 8 
5) = dab, = iil, eer ac, 9 = bd, 
where a, b,c, d are bs a prime (for a, 8, y, 6 are), and c and 
dare odd. Substituting in 2y?— 6?=0o?, we have 
a? (166? — c?) — 2ad . be + d? (ce? — b?) = 0. 
But as a/d is rational we have 
bc? + 16 (0? — ce) (BP —@) = sq. = 62, 
say. Putting 2b=7, c= &, this is &— 4£n?+ n= C2, where & is 
prime to 7, Also numerically 
En =2b0< 4abed < y?-—& <9 or < &, 
En < l6abed < B?—a? < 8? or < a”. 
* If it is B—a we put (8+a)/2=bd, (6- de 4ab, (y+4)/2=ac, (y— 5)/2=bd 
and get the same result as that given in the text. 
