| 
“ 
4 
a iI 
120 Mr Pocklington, Some Diophantine Impossibilities. 
w= Wy", y= 8", or if w is odd w+v=a", u—-v=P®, u=¥y 
vy =2"-1§", whence a+ 8” =(2y)" or a” — B™ = (26)” respectively. 
Hence 2%” + y= 2 is impossible for all values of n for which | 
a” +y"= 2" is impossible, that is according to a statement of 
‘Fermat *, for all values of n greater than 2, or according to a 
proof by Kummer, for all values of n that are divisible by any 
odd prime less than 100. Our equation is also impossible if n_ 
is even. 
14. We can of course apply the method of this paper to - 
a” + y” = 2°, but we get no results other than those found by 
Abelt. We notice, however, that if it has any solution it has 
one where x is prime to y. Hence (ayz)”=a"y"(a" + y”) is of 
the form w”=wuv(u+ v) with u prime to v,so that the problem 
reduces to that of proving that an nth power cannot be repre- 
sented primitively in a certain binary cubic form. We can now — 
apply the criterion that the number can be represented, but the 
resulting condition is so complex that further progress appears 
impossible. 
15. Summary. We have proved or given a method for 
proving that the following equations have no solutions in which 
“, Y, 2 are rational fractions or integers other than zero :— 
a*— py’ = 2, where p is a prime of the form 8m+ 3; a+ 2y3=27; 
a — 8yi= 2"; at — yt = pz, where p is a prime of the form 8m +3; 
(a? + y?)? — Na’y? = 2, where JN is odd, not of the form 8m+3 and 
not divisible by any prime of the form 4m+1, and V —4 is an 
odd power of a prime (including the case V =1); 
(a2 ae yy Ee Nay? = 2, 
where JV is odd, not of the form 8m+5 and not divisible by any 
prime of the form 4m+1, and V +4 is an odd power of a prime; 
(?+y?yP —2Na%y?= 2, where N is of the form 8m+7 and is 
divisible only by primes of the form 8m-+ 7, and V —2 is an odd 
power of a prime; (a+ y?)? + 2Na’y?= 2?, where NV + 2 is an odd 
power of a prime and WV 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; (#@+yP— 8Na*y?= 2, where NV is of the form 4m+3, 
is divisible only by primes of that form, and 2N —1 is an odd 
power of a prime; (a? +7) + 8Na*y?=2, where JN is of the form 
4m-+1 and divisible only by primes of the form 4m+3, and 
2N +1 is an odd power of a prime; at—na*y?+y!=2 for the - 
following values of m not included in the general results given 
above, n=17, 18, 20, 23, 24, 27; a+ na*y?+y*= 2" for the 
following values of n not included in the general results given 
* Diophantus, p.61. 
t+ See H. J..S. Smith, Brit. Assoc. Report, 1860 (Oxford), p. 151. 
£ N. H. Abel, Oeuvres Completes (Christiania, 1839), vol. 1. p. 264. 
