fe 
E . 
Mr Pocklington, Some Diophantine Impossibilities. 121 
above, n=15, 19, 22, 91; a2”+y"=2 for all values of n for 
which the equation a” +4” =z” has no solution in integers other 
than zero. We have also indicated the proof that the following 
equations have no solutions in which 2, y, z are rational fractions 
or integers other than zero, # and if being arithmetically unequal ; 
wht naty? +y=2 if n=7, 14; ~ ney? i gP Se it = Ms and 
proved the impossibility of ae by’)? + 128a°y? = 2? if a, y, "2 are 
rational fractions or integers other than zero and «fy #5y 
numerically. We have proved that four consecutive terms of an 
arithmetical progression cannot each be square, unless all are 
egual; and that the first, second, fifth and tenth terms cannot 
each be square unless the first is zero or all are equal. We have 
solved the equation a — 4a7y? + y'= 2 completely and given the 
nature of the complete solution of 2a*—yt= 2 and y* —2a* = 2, 
We have also shown that Fermat’s impossibility may be made to 
depend on the properties of a binary cubic form. 
