Just as an integer solution to a² + b2 = c2 describes a rectangle whose length, width, and diagonal all are integers, so an integer solution to a² +62 + c2 = d describes a three-dimensional rectangular box with integer dimensions and integer diagonal. Some such solutions can be found by combining results from the planar case; for instance, combining 32 + 42 = 52 and 52 + 122 + 132 yields 32 +42 + 122 = 132.
a. There are also solutions in which no two of the three dimensional measurements yield an integer diagonal in their plane, but the length of the main diagonal of the three-dimensional box is an in- teger. Find at least one of them.