And:
x² = 5486968438957476
2xy = 1828989479652492
33y² = 5029721069044353
sum = 12345678987654321
QED.
Obviously, y can also be negative, which will give:
Third possibility:
x = 12345679 – 86419753 = -74074074
Fourth possibility:
x = 12345679 + 86419753 = 98765432
FINALLY, we can conclude there are AT LEAST four more solutions beyond the trivial
(x; y) = (-111,111,111 ; 0)
(x; y) = (111,111,111 ; 0)
that are:
(x; y) = (-98,765,432 ; 12345679)
(x; y) = (74074074 ; 12345679)
(x; y) = (98,765,432 ; -12345679)
(x; y) = (-74074074 ; -12345679)
Note 1:
Because of the assumption made, we cannot be sure to have ALL the solutions.
Note 2:
I solved your problem assuming x and y were relative integers.
Obviously, if x and y are natural integers, you only have 2 solutions intead of 6.
Regards,
Dragon.Jade :-)