Proving that a list of perfect square numbers is complete
Solution 1:
$y^2=1+12x^2(1+x) \implies (12 y)^2 = (12 x)^3 + 12 (12 x)^2 + 144$
Magma code for positive $y$ only:
S:= IntegralPoints(EllipticCurve([0,12,0,0,144]));
for s in S do
x:= s[1]/12;
if x eq Floor(x) then
print "(",x,", ",Abs(s[2]/12),")";
end if;
end for;
Output:
( -1 , 1 )
( 0 , 1 )
( 1 , 5 )
( 4 , 31 )
( 6 , 55 )