Reporting on some computations. The : the only solutions to $F_k - F_j = x^2$ with $0\leq j < k$ and $1 \leq x \leq 10^5$ 10^6$ are these:
{0,1,1}, {0,2,1}, {1,3,1}, {2,3,1}, {3,4,1},
{1,5,2}, {2,5,2}, {5,8,4}, {6,11,9}, {0,12,12},
{11,13,12}, {13,14,12}, {6,13,15}, {9,15,24}
where the format is {j,k,x}.

