Mr Pocklington, Some Diophantine Impossibilities. 115 
and as (7+ y?)/2 is odd we have 27/4=5, mod. 8, which is im- 
possible. Hence the proposed equation is impossible. 
8. If possible let (a+ y+ Na’y?=2, where N is odd, not 
of the form 8n +5, and not divisible by any prime of the form 
4n+1, and N +4 is an odd power of a prime (it cannot be an 
even power, for it is odd and not =1, mod. 8). As before we take 
the solution for which zy is least, and see that « is prime to y and 
one of them even, say y, and that NV is prime to a+y, Hence 
e+ y=l—mv*, sy=2uv. As before the last gives «=a8, 
y = 278, w= ay, v= 86, where a and 8 are odd, a, B, y, 6 prime 
to each other in pairs, and J prime to 8 and 6, m prime to a and 
y. Substituting in the previous equation we have 
a? (Iy? — 2) = 8° (dey? + mB), 
The determinant of the coefficients in the brackets is 
Neate 4, Sor, 
say, and hence the greatest common divisor of the brackets is p’, 
where 7X + 2k+1. Proceeding as before we have ly?— 6 = p'o, 
4n? + mB? = po. First suppose that X is even. Then the first 
equation shows that J=1 (and so m=) and that y is odd. 
Hence as a and # are odd the second gives V+ 4=1, mod. 8, 
which is contrary to the hypothesis. Next suppose that is odd. 
Solving for 8 and y we find lo? — 48° = p*B?, a + md? = py’, where 
p=2k+1—X and is even. The first equation shows that /=1 
(and so m=) and then gives a= &+ 7°, 6=&, and on sub- 
stituting in the second we get (&?+ 7”)? + N&n? = p*y? = square. 
This equation is of the original form, and &)=6< y< zy, which 
is contrary to the assumption. Hence the equation in question 
is impossible. 
In a closely similar way we can prove the impossibility of 
(a? + y°?)? — 2Na*y? = 22, where WN is of the form 8m + 7, is divisible 
only by primes of the form 8m+ 7, and NV — 2 is an odd power of 
a prime; that of (@+7°)+2Nay?=2, where N+2 is an odd 
power of a prime and N is of the form 8m +1 and either divisible 
only by primes of the form 8m+3 or only by those of the form 
Sm+7; that of (2+7)?—8Na’y?=2, where WN is of the form 
4m, + 3, is divisible only by primes of that form, and 2 —1 is an 
odd power of a prime; and that of (a+y)? + 8Na*y’, where VV 
is of the form 4m +1, is divisible only by primes of the form 
4m +3, and 2N +1 is an odd power of a prime. (We easily see 
that in each case the power is necessarily odd.) 
9. The general method can also be applied to many other 
eases, the necessary exclusion of some alternatives being effected 
by noticing that an equation such as aa?+ 6?= 66" can only hold 
if some such condition as that —a@ and b are quadratic residues 
8—2 
