Oliver:

I am trying to reach the range of 2 to 1010 for the smallest side of 
the integer cuboid, and had about 1/2 of this range done.

Bill Butler did odd sides to 1010, but missed the even sides.

My search is exhaustive over that range.

Recently Clifford Reiter found perfect parallelepipeds, I am extending 
his table, and found 3 missing ones over his range (sorry to say)

They are harder to find than cuboids as the search space is O3 and I 
haven't found a way to shorten this.

Randall

P.S. one more thing, I provided Dr Richard K Guy over 60 references on D18 
problem

but recent results of the perfect parallelepiped shows a connection 
between heron triangles with rational medians, triangles with rational 
bisectors and square sides, and the perfect cuboid, all of which seem to 
point strongly that the perfect cuboid does NOT exist.