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Proof without words that the difference of two consecutive cubes is a centered hexagonal number by arranging n 3 semitransparent balls in a cube and viewing along a space diagonal – colour denotes cube layer and line style denotes hex number
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A Cabtaxi number is the smallest positive number that can be expressed as a sum of two integer cubes in n ways, allowing the cubes to be negative or zero as well as positive. The smallest cabtaxi number after Cabtaxi(1) = 0, is Cabtaxi(2) = 91, [ 5 ] expressed as:
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Below are identities whose left-hand sides are commonly used as patterns (this means that the variables E and F that appear in these identities may represent any subexpression of the expression that has to be factorized). [6] Visual proof of the differences between two squares and two cubes. Difference of two squares
The difference of two squares can also be illustrated geometrically as the difference of two square areas in a plane. In the diagram, the shaded part represents the difference between the areas of the two squares, i.e. a 2 − b 2 {\displaystyle a^{2}-b^{2}} .
A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. This shows that the square of the n th triangular number is equal to the sum of the first n cube numbers. Also, the square of the n th triangular number is the same as the sum of the cubes of the integers 1 to n.
1729 can be expressed as a sum of two positive cubes in two ways, illustrated geometrically. 1729 is also known as Ramanujan number or Hardy–Ramanujan number, named after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan who was ill in a hospital.