Visualising a Difference of Two Cubes

For some time I've thought about a visual representation of a positive integer that is the difference of two cubes. For example, the number 973 can be represented thus:$$ \begin{align} 973 &= 1000 - 27 \\ &=10^3-3^3 \end{align} $$Given the limitations of Nano Bananas, I thought that a rought sketch of how I wanted to represent the number would help. This is shown in Figure 1.

Figure 1

I uploaded this sketch to Gemini along with the following prompt:

Use the uploaded image to create a realistic depiction of the cube shown following the heavy lines that drift the edges. The large cube has a smaller cube removed from it. The cube sits on a flat surface and is golden coloured with soft lighting highlighting the missing section of the cube. Maintain two point perspective and incorporate the numerals shown.

Figure 2 shows the result:

Figure 2

I asked Gemini:

Given today's gold price what would a cube of 100% pure gold with side of 10 millimetres be worth in US dollars. Also price a cube of gold with side of 3 millimetres.

The response:

To calculate the value of both cubes, we need today's gold price and the physical density of pure gold.

As of February 24, 2026, the spot price for 24k (100% pure) gold is approximately 166.69 USD per gram. The density of pure gold is 19.32 grams per cubic centimeter (g/cm³).

1. The 10 mm Cube

Volume: 10 mm × 10 mm × 10 mm = 1,000 mm³, which is exactly 1 cm³.

Mass: 1 cm³ × 19.32 g/cm³ = 19.32 grams.

Value: 19.32 grams × 166.69 USD/gram = 3,220.45 USD.

2. The 3 mm Cube

Volume: 3 mm × 3 mm × 3 mm = 27 mm³, which is 0.027 cm³.

Mass: 0.027 cm³ × 19.32 g/cm³ = 0.52164 grams.

Value: 0.52164 grams × 166.69 USD/gram = 86.95 USD.

Connecting this back to the golden object depicted in your image—a 10 mm cube with a 3 mm cubed corner removed—its total gold value today would be 3,220.45 USD minus 86.95 USD, which equals 3,133.50 USD.

There are 824 numbers up to 40000 that are the difference of two cubes (some in more ways than one) but only those from 28000 to 40000 are listed below (permalink will generate the full list):

28063, 28415, 28460, 28519, 28568, 28656, 28672, 28701, 28737, 28791, 28828, 28854, 29051, 29062, 29078, 29080, 29107, 29279, 29393, 29402, 29448, 29528, 29575, 29617, 29666, 29701, 29727, 29735, 29763, 29764, 29783, 29790, 30016, 30024, 30043, 30105, 30248, 30301, 30312, 30483, 30571, 30708, 30807, 30907, 30970, 31024, 31031, 31040, 31085, 31106, 31213, 31228, 31232, 31304, 31437, 31519, 31768, 31806, 31841, 31869, 31976, 32039, 32137, 32227, 32256, 32319, 32425, 32445, 32464, 32465, 32552, 32562, 32643, 32704, 32741, 32760, 32761, 32767, 32832, 32851, 32858, 32920, 32949, 32984, 33077, 33193, 33336, 33391, 33472, 33614, 33724, 33740, 33752, 33875, 34027, 34047, 34209, 34391, 34489, 34531, 34606, 34658, 34669, 34784, 34875, 34902, 34930, 34937, 35008, 35028, 35163, 35189, 35208, 35315, 35317, 35425, 35576, 35594, 35721, 35812, 35873, 35910, 35929, 35936, 35971, 36008, 36016, 36153, 36253, 36297, 36316, 36504, 36506, 36560, 36631, 36632, 36785, 36829, 37000, 37043, 37107, 37296, 37297, 37367, 37395, 37448, 37449, 37576, 37648, 37962, 37969, 37973, 38017, 38142, 38151, 38285, 38304, 38402, 38486, 38528, 38575, 38619, 38647, 38656, 38779, 38792, 38961, 39004, 39088, 39130, 39179, 39240, 39247, 39277, 39296, 39303, 39331, 39368, 39500, 39611, 39636, 39797, 39807, 39815, 39816, 39823

Notice how there is clustering about cubic numbers. For example, \(31^3=32768\) and so the following are members of the sequence:

• \(31^3-1^3=32767\)
• \(31^3-2^3=32760\)
• \(31^3-3^3=32741\)
• \(31^3-4^3=32704\)

However, looking back at the range of numbers above, notice how 32761 slips in (to form a consecutive pair with 32760) and this is because:$$32761=105^3-104^3$$This number is also mentioned below. These numbers that are differences of two cubes form OEIS A181123. The OEIS comments are interesting:

Because \(x^3-y^3 = (x-y)(x^2+xy+y^2)\), the difference of two cubes is a prime number only if \(x=y+1\), in which case all the primes are cuban, see A002407.

The difference can be a square (see A038597), but Fermat's Last Theorem prevents the difference from ever being a cube.

The numbers that are square and the difference of two cubes are 169, 784, 2401, 10816, 21609, 32761 and 35721 with squares as shown:$$ \begin{align} 169 &=13^2 =8^3-7^3\\ 784 &= 28^2 =10^3-6^3\\ 2401 &=49^2=14^3-7^3 \\ 10816 &=104^2=32^3-28^3 \\ 21609 &= 147^2=28^3-7^3 \\32761 &=181^2 =105^3-104^3 \\35761 &= 189^2=33^3-6^3 \end{align}$$There are 44 cuban primes in the range and they are:

7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657, 1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447, 5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227, 27361, 33391, 35317

With all these cuban primes, the difference between the bases of the primes must be 1 e.g. \(7=2^3-1^3\).

In conclusion, the main point of this post was to show one way in which a difference of two cubes can be represented physically. While Mathematics is highly abstract, it's still possible on occasion to make it more concrete.