110 Mr Pocklington, Some Diophantine Impossibiltties. 
From this we easily solve completely the problem of finding the 
parallelograms that have their sides and diagonals integral. The 
solution is that the sides are a8—y6é and ay+6, and the 
diagonals a8+yd+ay— 8S and aB+yd—ay+ Bd. We may 
without loss of generality take a to be positive, and then in order 
that x, y, u,v may be positive and form a real parallelogram we — 
must have §, y, 6 positive, and a>6, B>y¥. 
The third lemma is that if «y= uv and @ is prime to v and y 
to u,thenv=+u, y=+v. For from the first condition x divides — 
u, and from the second w divides w. 5 
3. Consider a'— py!= 2°, where p is a prime of the form 
8m + 3, and suppose that we have that solution in which zy has 
the least value. Then # is prime to y. It is also prime to gp, for 
otherwise 2? is divisible by p and hence by p?, so that py* is” 
divisible by p®, which is impossible, as y, being prime to «, is not 
divisible by p. Also y is even, for otherwise we should have 
2=5 or 6, mod. 8. Hence a =w?+ pvt, y2=2uv, where wu is 
prime to pv and one of them is even. If wu is even, v is odd and 
x?= 38, mod. 4, which is impossible, so that v is even. Hence — 
+ u= &— pr’, v= 2m, where & is prime to py and one of them 
is even, which gives y?= + 4& (E2— py). Hence =a? 4 = 87, 
E?- pry?=+y or a'—pRt=+y*, and a or B is even. Hence} 
+ ?=1, mod. 4, so that the upper sign must be taken, and the 
equation is of the same form as that considered. Also 
CHISE 3 faipe< OPK GPU, 
that 1s we have found a solution in which zy has a value less than 
the least that it can have, and we infer that the equation con- 
sidered is impossible. In the same way we show that the equation 
x*+ 2y!= 2" is impossible, and the impossibility of a — 8yi'=2 
can be proved similarly. The complete solutions of a*— 2yt=2 
and 2a*— y*= 2? can be found. 
4. The equation «+ y*= nz’ is impossible if n (supposed not 
divisible by any square) contains an odd prime not of the form 
8m + 1, for if it has a solution it has one in which « is prime to y, 
and a+ y* is then not divisible by any odd prime that is not of 
this form. If n=17 there is a solution =2, y=1 and we can 
prove by Fermat’s* method that the number of distinct solutions 
1s infinite. 
If possible let a, y, z be that solution of the equation 
ah yt = p2’, 
where p is a prime of the form 8m +3, in which wy has the least 
value. Then as before # is prime to pz and z is even. Hence 
* «Doctrinae Analyticae Inventum novum ” (prefixed to S. Fermat’s edition of 
Diophantus), p. 26 et seq. 
