Taxicab Numbers and Cube Calculations

Let's dive into the fascinating world of Taxicab numbers and cube calculations. We will work through problem three regarding sums of cubes and problem four involving basic cube evaluations.

Starting with problem 3 a, we need to verify the famous Hardy Ramanujan number, one thousand seven hundred and twenty nine, is the sum of two different pairs of cubes.

Both calculations give one thousand seven hundred and twenty nine, so the verification is complete.

Now for problem 3 b, let's express four thousand one hundred and four as the sum of two cubes in two different ways. Let's list some cubes near this value.

Notice that four thousand ninety six is very close to four thousand one hundred and four. The difference is eight, which is two cubed.

Next, let's try the cube of fifteen, which is three thousand three hundred seventy five. The difference is seven hundred twenty nine, which is exactly nine cubed.

In part c, we check if one thousand can be expressed as a sum of two positive cubes. We already know one thousand is ten cubed.